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GENERAL COMMENTARY article

Front. Phys., 30 April 2024
Sec. Atomic and Molecular Physics

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

  • Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA), La Plata, Argentina

A Commentary on
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, Vr0= and V(r) exhibits a minimum value at the interatomic equilibrium distance re given by Vre=0. The dissociation energy is defined as De=VrVre, assuming that Vr is finite. One of the simplest examples is given by the Kratzer potential

VKDe,re,r=2Dererre22r2(1)

that supports an infinite number of bound states with energies EvjKDe,re, 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,

Vr=2Dererre22r22λ+γeαr,(2)

where λ and γ are “shifting parameters” that can be “adjusted as desired.” We should add, provided that 2λ + γeαr > 0 for all r. They claimed that De and re 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 Enl, where n and l are “the radial and orbital angular momentum quantum numbers,” respectively. According to Ibrahim et al., n = 0, 1, …, Nmax, where “Nmax is the maximum vibration quantum number.”

One may think that this model offers five adjustable parameters: De, re, λ, γ, and α. However, if we rewrite it as

Vr=2Dererre22r21+γeαr,De=2λDe,γ=γ2λ,(3)

we realize that there are just four independent parameters: De, re, 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

VK2λ+γDe,re,r<Vr<VK2λDe,re,r,(4)

then we conclude that the true eigenvalues EvJ of the Schrödinger equation with the potential (2) are bounded as

EvjK2λ+γDe,re<EvJ<EvjK2λDe,re.(5)

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.

1rα1eαr(6)

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

Vre=λαeαre,(7)

then re is not the equilibrium bond length, and consequently, De 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 Enl for the molecules H2, LiH, HCl, and CO with respect to De, re, 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 Enl for CO for different values of De because only one value of the dissociation energy is compatible with that molecule.

We can derive an approximate expression for re in terms of the true equilibrium bond length r̃e for the potential (2). If we solve Vr̃e=0 for re by means of a power series, we obtain

re=r̃e+αγr̃e22γ+2λα2γr̃e3γ+4λ4γ+2λ2+α3γr̃e4γ2+4γλ+8λ28γ+2λ3+.(8)

Therefore, if we insert the experimental equilibrium bond length r̃e and the other model parameters, we obtain re appearing in the potential (2). Note that re=r̃e when either 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 De, 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

Zβ,Nmax=n=0NmaxeβEnl,(9)

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 De and re 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

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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

CrossRef Full Text | Google Scholar

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

Google Scholar

3. McQuarrie DA. Statistical mechanics. New York: Harper and Row (1976).

Google Scholar

4. Fernández FM. On the application of statistical mechanics to diatomic molecules (2021). arXiv:2112.10010 [physics.chem-ph].

Google Scholar

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 States

Reviewed by:

Jayanta Kumar Saha, Aliah University, India

Copyright © 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

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.