Comment on "The Rotation-Vibration Spectrum of Diatomic Molecules with the Tietz-Hua Rotating Oscillator"

TL;DR

This paper critiques the application of the Nikiforov-Uvarov method to the Tietz-Hua potential, clarifying valid parameter ranges and correcting previous numerical results for diatomic molecules, emphasizing the need for hypergeometric functions in solutions.

Contribution

It establishes the correct parameter conditions for applying the Nikiforov-Uvarov method to the Tietz-Hua potential and corrects prior numerical inaccuracies in molecular energy calculations.

Findings

The method is valid only when $e^{-b_{h}r_{e}}\leq c_{h}<1$.

Previous numerical results for certain molecules are incorrect.

Solutions involve hypergeometric functions and require solving transcendentally.

Abstract

We present arguments demonstrating that the application of the Nikiforov-Uvarov polynomial method to solve the Schr\"odinger equation with the Tietz-Hua potential is valid only when $e^{-b_{h}r_{e}}\leq c_{h}<1$ and $r_{0}<r<+\infty $. In particular, it is pointed out that the numerical results with $c_{h}\neq 0$ for the diatomic molecules $\mathrm{HF}$, $\mathrm{N}_{2}$, $\mathrm{I}_{2}$, $\mathrm{H}_{2}$, $\mathrm{O}_{2}$ and $\mathrm{O}_{2}^{+}$ given in Tables 3-5 by Hamzavi and co-workers are wrong. When $-1\leq c_{h}<0$ or $0<c_{h}<e^{-b_{h}r_{e}}$, this approach is not suitable. In both cases, it is shown that the solutions of the Schr\"odinger equation are expressed in terms of the generalized hypergeometric functions $_{2}F_{1}\left( a,b,c;z\right) $. The determination of the energy levels requires the solution of transcendantal equations involving the hypergeometric function by means of the numerical procedure.