On the one-dimensional Morse potential as limit of a class of multiparameter exponential-type radial potential

Despite their differences, the exponential-type radial potential and the Morse potential can be compared not only analytically but also through spectroscopic calculations. In this work, the eigensolutions of the Schrödinger equation coming from the hypergeometric differential equation are used to show the relationship that exist between the Morse potential and a class of multiparametric exponential-type radial (MER) potential. To do that, we rewrite the MER potential and its corresponding energy spectrum such that their main parameters as its depth V₀, the equilibrium point r_e and its width parameter β explicitly appear in the potential. After, through the infinity limit applied to the r_e parameter we show that the MER potential tends to the Morse potential. Likewise, under the same limit, the energy spectrum and the number of bound states tend to those of the Morse potential which indicates the equivalence of using the Morse potential or MER potentials in the calculation of spectroscopic transitions for diatomic molecules. As a useful application, we select particular values of the involved parameters in the MER potential in such a way that leads to the Poschl–Teller II potential. Thus, after apply the aforementioned limit to this potential, it is shown that energy spectrum and number of states correspond to those of Morse potential. Although the usefulness of the proposal is exemplified in the case of the molecule HCl³⁵ and the radial Posch–Teller potential, the method can be straightforwardly used with other specific exponential-type radial potentials as well as with other diatomic molecules.