Non-relativistic energy equations for diatomic molecules constrained in a deformed hyperbolic potential function

In this paper, the approximate analytical energy equations for the deformed hyperbolic potential have been obtained for arbitrary parameters of the potential. The potential function was transformed to a molecular potential by subjecting it to the Varshni conditions which allows for the determination of the energy levels of diatomic molecules. The molecular vibrational energy spectra for \(L{i}_{2}(G{}_{\hspace{0.17em}}{}^{1}\Pi {\hspace{0.17em}}_{g})\) $L{i}_2(G{}_{}{}^1\mathrm{\Pi }{}_g)$ , \({N}_{2}({X}^{1}{\sum }_{g}^{+}\hspace{0.17em})\) $N_2(X^1{\sum }_g^{+})$ , \(RbH({X}^{1}{\sum \hspace{0.17em}}^{+})\) $RbH(X^1{\sum }^{+})$ , \(NI({b}^{1}{\sum \hspace{0.17em}}^{+})\) $NI(b^1{\sum }^{+})$ , and \(BrF({X}^{1}{\sum \hspace{0.17em}}^{+})\) $BrF(X^1{\sum }^{+})$ diatomic molecules were obtained and found to match with the results obtained with another analytical approach, potential functions, and experimental data. The noticeable slight differences in the approximate energy spectra obtained in this work and existing literature may be ascribed to the analytical method, computational approach, and the accuracy of the molecular potential functions. The obtained energy equations were used to determine the energy of a particle for arbitrary parameters of the potential function. The obtained energy is bounded and increases with the increase in the quantum numbers. The results conformed to the ones obtained via the path integral approach and numerical solutions obtained via the MATHEMATICA program.

The energy spectra equations were obtained via the Nikiforov-Uvarov approach and semi-classical WKB approximation. The Pekeris approximation has been applied to resolve the difficulty in solving the complete energy spectrum of the non-relativistic wave equation for the potential function. The numerical data of the energy spectra was obtained using the MATHEMATICA program.