Solutions of the Schrodinger equation of the shifted screened Kratzer potential and its thermodynamic functions using the extended Nikiforov–Uvarov method

* by Ibrahim N, Okorie US, Sulaiman N, Rampho GJ and Ramantswana M (2022) Solutions of the Schrodinger equation of the shifted screened Kratzer potential and its thermodynamic functions using the extended Nikiforov–Uvarov method. Front. Phys. 10:988279. doi: 10.3389/fphy.2022.988279*

## 1 Introduction

In a paper presented recently in this journal, Ibrahim et al. [1] proposed the shifted screened Kratzer potential for the analysis of the spectra of diatomic molecules. After a suitable transformation of part of the potential energy function, they derived analytical expressions for the vibration–rotation eigenvalues by means of the Nikiforov–Uvarov method. Ibrahim et al. also calculated the canonical partition function and some thermodynamic functions. The purpose of this commentary is the analysis of their results. In Section 2, we summarize some relevant aspects of the potential energy function for a diatomic molecule and analyze the spectrum of the shifted screened Kratzer potential. We also discuss the calculation of the canonical partition function. Finally, in Section 3, we summarize the main results and draw conclusions.

## 2 The shifted screened Kratzer potential

We begin the discussion summarizing a few well-known properties of the interatomic potential *V*(*r*) for a diatomic molecule. Commonly, *V*(*r*) exhibits a minimum value at the interatomic equilibrium distance *r*_{e} given by

that supports an infinite number of bound states with energies *v* = 0, 1, …, and *J* = 0, 1, …, where *v* and *J* are the vibration and rotation quantum numbers, respectively. The Schrödinger equation with the Kratzer potential is exactly solvable.

Ibrahim et al. [1] proposed the so-called shifted screened Kratzer potential,

where *λ* and *γ* are “shifting parameters” that can be “adjusted as desired.” We should add, provided that 2*λ* + *γe*^{−αr} > 0 for all *r*. They claimed that *D*_{e} and *r*_{e} are the dissociation energy and equilibrium bond length, respectively (we will discuss this point further). After a suitable transformation of *r*^{−1} and *r*^{−2} and further application of the Nikiforov–Uvarov method, they obtained the eigenvalues *E*_{nl}, where *n* and *l* are “the radial and orbital angular momentum quantum numbers,” respectively. According to Ibrahim et al., *n* = 0, 1, …, *N*_{max}, where “*N*_{max} is the maximum vibration quantum number.”

One may think that this model offers five adjustable parameters: *D*_{e}, *r*_{e}, *λ*, *γ*, and *α*. However, if we rewrite it as

we realize that there are just four independent parameters: *r*_{e}, *γ*′, and *α*.

We can easily derive some properties of the spectrum supported by the potential (2) without solving the Schrödinger equation. If we take into account that

then we conclude that the true eigenvalues *E*_{vJ} of the Schrödinger equation with the potential (2) are bounded as

Since both the lower and upper bounds correspond to the infinite spectra of Kratzer potentials, we conclude that the potential (2) also supports an infinite number of states with negative energy eigenvalues. One may wonder why Ibrahim et al. obtained a finite number of bound states instead of an infinite number of them. The reason is the transformation.

that is reasonable when *r* is close to the origin, but it completely changes the behavior of the potential at infinity. The infinite spectrum of *V*(*r*) is due to the behavior of the potential as *r*^{−1} for large values of *r*, while the transformed potential exhibits a completely different behavior.

One may argue that the approximation (6) leads to reasonably accurate eigenvalues for small values of the vibration quantum number. However, the approach followed by Ibrahim et al. faces another serious shortcoming. Since

then *r*_{e} is not the equilibrium bond length, and consequently, *D*_{e} is not the dissociation energy. If, as it appears to be, Ibrahim et al. substituted experimental values for these parameters, the resulting curves are unsuitable for a physical application. There is something even more serious in the authors’ paper. Figures 1–4 show the variation in *E*_{nl} for the molecules *H*_{2}, *LiH*, *HCl*, and *CO* with respect to *D*_{e}, *r*_{e}, *α*, and *μ*, respectively. Such an analysis is incomprehensible because a given set of those parameters defines one and only one molecule. In other words, it is incorrect, for example, to show *E*_{nl} for *CO* for different values of *D*_{e} because only one value of the dissociation energy is compatible with that molecule.

We can derive an approximate expression for *r*_{e} in terms of the true equilibrium bond length *r*_{e} by means of a power series, we obtain

Therefore, if we insert the experimental equilibrium bond length *r*_{e} appearing in the potential (2). Note that *α* = 0 or *γ* = 0 because in such cases, *V*(*r*) becomes a Kratzer potential (see also Eq. 7). We may try a similar calculation for *D*_{e}, but it is not necessary to spend time in this kind of discussion because any book on spectroscopy shows that the standard procedure for the construction of a potential energy curve is to fit the theoretical spectrum to the experimental one [2].

In conclusion, we mention another weakness in the paper by Ibrahim et al. They calculated the canonical partition function as

overlooking the sum over the rotation quantum number *l* (commonly called *J* by spectroscopists and other scientists) [2, 3]. Consequently, the resulting expression for *Z* and the thermodynamic functions derived from it are unsuitable for the treatment of a physical problem. There are many examples of suitable physical applications of statistical mechanics in most textbooks on the subject [3] (see also a recent critique of a paper on the same subject and the example therein [4]).

## 3 Conclusion

Ibrahim et al. [1] proposed a potential for diatomic molecules of dubious utility [2]. The parameters *D*_{e} and *r*_{e} are not the dissociation energy and the equilibrium bond length, respectively, as they argued. The figures in their paper are incomprehensible. Finally, since they did not include the rotation quantum number in the calculation of the canonical partition function, the resulting thermodynamic functions are inapplicable to a physical system.

## Author contributions

FF: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing.

## Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

## Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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## References

1. Ibrahim N, Okorie US, Sulaiman N, Rampho GJ, Ramantswana M. Solutions of the Schrödinger equation of the shifted screened Kratzer potential and its thermodynamic functions using the extended Nikiforov- Uvarov method. *Front Phys* (2022) 10:988279. doi:10.3389/fphy.2022.988279

2. Herzberg G. *Molecular spectra and molecular structure. I. Spectra of diatomic molecules*. Second. New York: Van Nostrand Reinhold (1950).

Keywords: Kratzer potential, screened Kratzer potential, dissociation energy, equilibrium bond length, spectrum, partition function, thermodynamic functions

Citation: Fernández FM (2024) Commentary: Solutions of the Schrodinger equation of the shifted screened Kratzer potential and its thermodynamic functions using the extended Nikiforov–Uvarov method. *Front. Phys.* 12:1361070. doi: 10.3389/fphy.2024.1361070

Received: 24 December 2023; Accepted: 08 April 2024;

Published: 30 April 2024.

Edited by:

Han Pu, Rice University, United StatesReviewed by:

Jayanta Kumar Saha, Aliah University, IndiaCopyright © 2024 Fernández. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Francisco M. Fernández, fernande@quimica.unlp.edu.ar