Generalised tanh-shaped hyperbolic potential: Klein-Gordon equation's bound state solution

<jats:title>Abstract</jats:title> <jats:p>The development of potential theory heightens the understanding of fundamental interactions in quantum systems. In this paper, the bound state solution of the modified radial Klein-Gordon equation is presented for generalised tanh-shaped hyperbolic potential from the Nikiforov-Uvarov method. The resulting energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary $l$ states. It is also demonstrated that energy eigenvalues strongly correlate with potential parameters for quantum states. Considering particular cases, the generalised tanh-shaped hyperbolic potential and its derived energy eigenvalues exhibit good agreement with the reported findings. Furthermore, the rovibrational energies are calculated for three representative diatomic molecules, namely $\rm{H_{2}}$, $\rm{HCl}$ and $\rm{O_{2}}$. The lowest excitation energies are in perfect agreement with experimental results. Overall, the potential model is displayed to be a viable candidate for concurrently prescribing numerous quantum systems.</jats:p>