Energy and thermal magnetization of diatomic molecules under the enhanced Pöschl-Teller potential

Eyube, E. S.; Kamaludeen, M.; Vijinti, F. C.; Fwangle, I. I.; Isa, M. F.

doi:10.1038/s41598-026-49748-w

Download PDF

Article
Open access
Published: 25 April 2026

Energy and thermal magnetization of diatomic molecules under the enhanced Pöschl-Teller potential

E. S. Eyube ORCID: orcid.org/0000-0001-5059-0393¹,
M. Kamaludeen¹,
F. C. Vijinti²,
I. I. Fwangle³ &
…
M. F. Isa¹

Scientific Reports , Article number: (2026) Cite this article

488 Accesses
Metrics details

We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors present which affect the content, and all legal disclaimers apply.

Subjects

Abstract

The Schrödinger equation is solved analytically for a particle subject to an enhanced Pöschl-Teller (EPT) potential in the presence of an external magnetic field and an Aharonov-Bohm flux field that induces topological phase effects. The bound state ro-vibrational energy equations are obtained using the generalized fractional Nikiforov-Uvarov method in conjunction with the Pekeris approximation scheme. The fractional order is introduced as an effective parameter that accounts for nonlocal and anharmonic ro-vibrational interactions that are not fully represented within the standard integer order framework. Based on these energy expressions, the mean thermal magnetization is derived within the partition function formalism. Numerical applications to diatomic molecules such as CO (X ¹Σ⁺), Cs₂ (3 ³Σ_g⁺), ICl (X ¹Σ_g⁺), ⁷Li₂ (1 ³Δ_g), Na₂ (C(2) ¹Π_u), and NaK (c ³Σ⁺) show that the mean percentage absolute deviation values decrease from 0.0990%, 0.1407%, 1.0027%, 1.9504%, 0.1234%, and 0.9811% to 0.0905%, 0.1020%, 0.5501%, 1.1756%, 0.0474%, and 0.4840% when fractional parameters are incorporated, indicating improved agreement with experimental data and enhanced flexibility of the ro-vibrational energy model. The analysis further shows that, at fixed temperatures, the mean thermal magnetization of the Na₂ (C(2) ¹Π_u) dimer increases with increasing magnetic field strength, highlighting the sensitivity of the system to external field variations. These results establish the EPT potential combined with a fractional order formulation as a reliable and adaptable analytical framework for describing the quantum and thermomagnetic behavior of diatomic molecules influenced by magnetic and topological quantum fields.

Improved modelling for vibrational energies of diatomic molecules using the generalized fractional derivative

Article Open access 09 April 2026

To study the thermodynamic properties of magnetic field on water based CoFe₂O₄ nanofluids

Article Open access 19 February 2025

Magneto-transport and Thermal properties of TiH diatomic molecule under the influence of magnetic and Aharonov-Bohm (AB) fields

Article Open access 14 September 2022

Author information

Authors and Affiliations

Department of Physics, Faculty of Physical Sciences, Modibbo Adama University, P.M.B. 2076, Yola, Adamawa State, Nigeria
E. S. Eyube, M. Kamaludeen & M. F. Isa
Department of Basic and Applied Science, College of Agriculture, Science and Technology, P.M.B. 1025, Jalingo, Taraba State, Nigeria
F. C. Vijinti
Department of Physical Science Education, Faculty of Education, Modibbo Adama University, P.M.B. 2076, Yola, Adamawa State, Nigeria
I. I. Fwangle

Authors

E. S. Eyube
View author publications
Search author on:PubMed Google Scholar
M. Kamaludeen
View author publications
Search author on:PubMed Google Scholar
F. C. Vijinti
View author publications
Search author on:PubMed Google Scholar
I. I. Fwangle
View author publications
Search author on:PubMed Google Scholar
M. F. Isa
View author publications
Search author on:PubMed Google Scholar

Corresponding author

Correspondence to E. S. Eyube.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

This appendix presents the derivation of the expansion coefficients κ_k and λ_k for k = 0, 1, 2. For brevity, approximations (6) and (7) are restated in the compact form.

$${\text{f}} \approx {\lambda _0}+{\lambda _1}{\text{u}}+{\lambda _2}{\text{v}}$$

(A1)

$${\text{g}} \approx {\kappa _0}+{\kappa _1}{\text{u}}+{\kappa _2}{\text{v}}$$

(A2)

where the associated definitions are

$${\text{f}}=\frac{{\rho _{\text{e} }^{2}}}{{{\rho ^2}}}$$

(A3)

$${\text{g}}=\frac{{{\rho _\text{e} }}}{\rho }\tanh \left( {\tfrac{1}{2}\alpha \rho } \right)$$

(A4)

$${\text{u}}={\text{csch} ^2}\alpha \left( {\rho - {\rho _0}} \right)$$

(A5)

$${\text{v}}={\text{csch} ^2}\alpha \left( {\rho - {\rho _0}} \right)\cosh \alpha \left( {\rho - {\rho _0}} \right)$$

(A6)

The objective is to expand the functions f(ρ), g(ρ), u(ρ), and v(ρ) in a Taylor series around the equilibrium point ρ = ρ_e, giving

$${\text{f}} \approx {{\text{f}}_{\text{e}}}+{{{\text{f}^{\prime}}}_{\text{e}}} \cdot \left( {\rho - {\rho _\text{e} }} \right)+\frac{1}{2}{{{\text{f}^{\prime\prime}}}_{\text{e}}} \cdot {\left( {\rho - {\rho _\text{e} }} \right)^2}+ \cdots$$

(A7)

$${\text{g}} \approx {{\text{g}}_{\text{e}}}+{{{\text{g}^{\prime}}}_{\text{e}}} \cdot \left( {\rho - {\rho _\text{e} }} \right)+\frac{1}{2}{{{\text{g}^{\prime\prime}}}_{\text{e}}} \cdot {\left( {\rho - {\rho _\text{e} }} \right)^2}+ \cdots$$

(A8)

$${\text{u}} \approx {{\text{u}}_{\text{e}}}+{{{\text{u}^{\prime}}}_{\text{e}}} \cdot \left( {\rho - {\rho _\text{e} }} \right)+\frac{1}{2}{{{\text{u}^{\prime\prime}}}_{\text{e}}} \cdot {\left( {\rho - {\rho _\text{e} }} \right)^2}+ \cdots$$

(A9)

$${\text{v}} \approx {{\text{v}}_{\text{e}}}+{{{\text{v}^{\prime}}}_{\text{e}}} \cdot \left( {\rho - {\rho _\text{e} }} \right)+\frac{1}{2}{{{\text{v}^{\prime\prime}}}_{\text{e}}} \cdot {\left( {\rho - {\rho _\text{e} }} \right)^2}+ \cdots$$

(A10)

Here, a prime denotes differentiation with respect to ρ. The symbols f_e, …, etc., represent the values of the corresponding functions evaluated at ρ = ρ_e. Using (A3) -(A6), the following expressions are obtained

$${{\text{f}}_{\text{e}}}=1,\quad {{{\text{f}^{\prime}}}_{\text{e}}}= - \frac{2}{{{\rho _\text{e} }}},\quad {{{\text{f}^{\prime\prime}}}_{\text{e}}}=\frac{6}{{\rho _{\text{e} }^{2}}}$$

(A11)

$${{\text{g}}_{\text{e}}}=\tanh \left( {\tfrac{1}{2}p} \right),\;{{{\text{g}^{\prime}}}_{\text{e}}}={{\text{g}}_{\text{e}}}\left( {\alpha \text{sech} p - \frac{1}{{{\rho _\text{e} }}}} \right),\;{{{\text{g}^{\prime\prime}}}_{\text{e}}}={{\text{g}}_{\text{e}}}\left\{ {\frac{1}{2}{\alpha ^2}{{\tanh }^2}\left( {\tfrac{1}{2}p} \right) - \frac{{2\alpha }}{{{\rho _\text{e} }}}\text{csch} p+\frac{2}{{\rho _{\text{e} }^{2}}}} \right\}$$

(A12)

$${{\text{u}}_{\text{e}}}={\text{csch} ^2}q,\quad {{{\text{u}^{\prime}}}_{\text{e}}}= - 2\alpha {{\text{u}}_{\text{e}}}\coth q,\quad {{{\text{u}^{\prime\prime}}}_{\text{e}}}=2{\alpha ^2}{{\text{u}}_{\text{e}}}\left( {2{{\coth }^2}q+{{\text{csch} }^2}q} \right)$$

(A13)

$${{\text{v}}_{{e}}}={\text{csch} ^2}q\cosh q,\quad {{{\text{v}^{\prime}}}_{\text{e}}}= - \alpha {{\text{v}}_{\text{e}}}\left( {2\coth q - \tanh q} \right),\quad {{{\text{v}^{\prime\prime}}}_{\text{e}}}={\alpha ^2}{{\text{v}}_{\text{e}}}\left( {{{\coth }^2}q+5{{\text{csch} }^2}q} \right)$$

(A14)

r$p=\alpha {\rho _{e} }$The auxiliary quantities $p=\alpha {\rho _\text{e} }$, and $q=\alpha \left( {{\rho _\text{e} } - {\rho _0}} \right)$ are introduced for notational simplicity. Substituting expansions (A7), (A9), and (A10) into (A1), and equating coefficients of ${\left( {{\rho _\text{e} } - {\rho _0}} \right)^0}$, ${\left( {{\rho _\text{e} } - {\rho _0}} \right)^1}$, and ${\left( {{\rho _\text{e} } - {\rho _0}} \right)^2}$, results in the linear system

$${\lambda _0}+{{\text{u}}_\text{e} }{\lambda _1}+{{\text{v}}_{\text{e}}}{\lambda _2}={{\text{f}}_{\text{e}}}$$

(A15)

$${{{\text{u}^{\prime}}}_{e} }{\lambda _1}+{{{\text{v}^{\prime}}}_{\text{e}}}{\lambda _2}={{{\text{f}^{\prime}}}_{\text{e}}}$$

(A16)

$${{{\text{u}^{\prime\prime}}}_\text{e} }{\lambda _1}+{{{\text{v}^{\prime\prime}}}_{\text{e}}}{\lambda _2}={{{\text{f}^{\prime\prime}}}_{\text{e}}}$$

(A17)

Solving (A15) -(A17) yields

$${\lambda _2}=\frac{{{{{{\text{u}^{\prime}}}}_{\text{e}}}{{{{\text{f}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{u}^{\prime\prime}}}}_{\text{e}}}{{{{\text{f}^{\prime}}}}_{\text{e}}}}}{{{{{{\text{u}^{\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{u}^{\prime\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime}}}}_{\text{e}}}}}=\frac{{2{{\cosh }^2}q+1}}{{p\sinh q}} - \frac{{3\cosh q}}{{{p^2}}}$$

(A18)

$${\lambda _1}=\frac{{{{{{\text{f}^{\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{f}^{\prime\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime}}}}_{\text{e}}}}}{{{{{{\text{u}^{\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{u}^{\prime\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime}}}}_{\text{e}}}}}= - \frac{{{{\cosh }^2}q+5}}{{2p\tanh q}}+\frac{{6{{\cosh }^2}q+3}}{{2{p^2}}}$$

(A19)

$${\lambda _0}={{\text{f}}_{\text{e}}} - {{\text{u}}_\text{e} }{\lambda _1} - {{\text{v}}_{\text{e}}}{\lambda _2}=1 - \frac{{3\coth q}}{{2p}}+\frac{3}{{2{p^2}}}$$

(A20)

To determine the general expressions for the coefficients κ_k (k = 0, 1, 2), expansions (A8)-(A10) are substituted into (A2), and coefficients of ${\left( {{\rho _\text{e} } - {\rho _0}} \right)^0}$, ${\left( {{\rho _\text{e} } - {\rho _0}} \right)^1}$, and ${\left( {{\rho _\text{e} } - {\rho _0}} \right)^2}$ are matched on both sides, giving

$${\kappa _0}+{{\text{u}}_{e} }{\kappa _1}+{{\text{v}}_{\text{e}}}{\kappa _2}={{\text{g}}_{\text{e}}}$$

(A21)

$${{{u^{\prime}}}_{e} }{\kappa _1}+{{{v^{\prime}}}_{\text{e}}}{\kappa _2}={{{g^{\prime}}}_{\text{e}}}$$

(A22)

$${{{u^{\prime\prime}}}_{e} }{\kappa _1}+{{{v^{\prime\prime}}}_{\text{e}}}{\kappa _2}={{{g^{\prime\prime}}}_{\text{e}}}$$

(A23)

Solving (A21) -(A23) gives the expansion coefficients

$${\kappa _2}=\frac{{{{{{\text{u}^{\prime}}}}_{\text{e}}}{{{{\text{g}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{u}^{\prime\prime}}}}_{\text{e}}}{{\text{g}^{\prime}}_{\text{e}}}}}{{{{{{\text{u}^{\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{u}^{\prime\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime}}}}_{\text{e}}}}}={\kappa _a}\left( {3\coth q\cosh q - \sinh q} \right)+{\kappa _b}\cosh q$$

(A24)

$${\kappa _1}=\frac{{{{{{\text{g}^{\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{g}^{\prime\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime}}}}_{\text{e}}}}}{{{{{{\text{u}^{\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime\prime}}}}_{\text{e}}} - {{{{\text{u}^{\prime\prime}}}}_{\text{e}}}{{{{\text{v}^{\prime}}}}_{\text{e}}}}}= - {\kappa _a}\left( {3\coth q{{\cosh }^2}q - \frac{5}{4}\sinh 2q} \right)+\frac{1}{2}{\kappa _b}\left( {{{\cosh }^2}q+1} \right)$$

(A25)

$${\kappa _0}=\tanh \left( {\tfrac{1}{2}p} \right) - \frac{3}{2}{\kappa _a}\coth q - {\kappa _b}\left( {2{{\coth }^2}q - \frac{1}{2}} \right)$$

(A26)

where

$${\kappa _a}=\frac{1}{2}{{sech} ^2}\left( {\tfrac{1}{2}p} \right) - \frac{{\tanh \left( {\tfrac{1}{2}p} \right)}}{p}$$

(A27)

$${\kappa _b}= - \frac{1}{2}{{sech} ^2}\left( {\tfrac{1}{2}p} \right)\tanh \left( {\tfrac{1}{2}p} \right) - \frac{{{{{sech} }^2}\left( {\tfrac{1}{2}p} \right)}}{p}+\frac{{2\tanh \left( {\tfrac{1}{2}p} \right)}}{{{p^2}}}$$

(A28)

Appendix B

This appendix provides the detailed derivation of the potential parameters α, q, U₀, U₁, and U₂. For convenience, the EPT potential defined in Eq. (2) is rewritten in the more tractable form.

$${\text{U}}\left( \rho \right)={{\text{U}}_0}+\frac{1}{{{y^2}}}{{\text{U}}_1} - \frac{x}{{{y^2}}}{{\text{U}}_2}$$

(B1)

where the auxiliary functions are

$$x=\cosh \alpha \left( {\rho - {\rho _0}} \right)$$

(B2)

$$y=\sinh \alpha \left( {\rho - {\rho _0}} \right)$$

(B3)

Equations (B1) -(B3) are connected through the identity ${x^2} - {y^2}=1$, which is used repeatedly in subsequent simplifications. To enforce the boundary condition (29a), note that in the limit ρ → ∞, both x → ∞ and y → ∞. Consequently, 1/y² → 0, x/y² → 0, and applying these limits to (B1) gives

$$\mathop {\lim }\limits_{{\rho \to \infty }} {\text{U}}\left( \rho \right)={{\text{U}}_0}$$

(B4)

At the equilibrium point ρ = ρ_e, substituting (B2) and (B3) into (B1) yields

$${\text{U}}\left( {{\rho _\text{e} }} \right)={{\text{U}}_0}+\frac{1}{{y_{\text{e} }^{2}}}{{\text{U}}_1} - \frac{{{x_\text{e} }}}{{y_{\text{e} }^{2}}}{{\text{U}}_2}$$

(B5)

where

$${x_\text{e} }=\cosh \alpha \left( {{\rho _\text{e} } - {\rho _0}} \right)$$

(B6)

$${y_\text{e} }=\sinh \alpha \left( {{\rho _\text{e} } - {\rho _0}} \right)$$

(B7)

Inserting (B4) and (B5) into condition (29a) leads to

$$- {{\text{U}}_1}+{x_\text{e} }{{\text{U}}_2}=y_{\text{e} }^{2}{D_\text{e} }$$

(B8)

To satisfy the derivative-based conditions (29b) -(29d), direct differentiation of (2) with respect to ρ is algebraically unwieldy. A more manageable approach is to work from the total differential of U(ρ)

$$\frac{{{\text{dU}}}}{{{\text{d}}\rho }}=\frac{{\partial {\text{U}}}}{{\partial x}}\frac{{dx}}{{{\text{d}}\rho }}+\frac{{\partial {\text{U}}}}{{\partial y}}\frac{{dy}}{{{\text{d}}\rho }}$$

(B9)

Differentiating (B2) and (B3) gives

$$\frac{{{\text{d}}x}}{{{\text{d}}\rho }}=\alpha \sinh \alpha \left( {\rho - {\rho _0}} \right) \equiv \alpha y$$

(B10)

$$\frac{{{\text{d}}y}}{{{\text{d}}\rho }}=\alpha \cosh \alpha \left( {\rho - {\rho _0}} \right) \equiv \alpha x$$

(B11)

Substituting (B10) and (B11) into (B9) yields the differential operator used to generate successive derivatives of the potential

$$\frac{{\text{d}}}{{{\text{d}}\rho }}=\alpha \left( {y\frac{\partial }{{\partial x}}+x\frac{\partial }{{\partial y}}} \right)$$

(B12)

Applying this operator to (B1) produces the first derivative

$${{\text{U}^{\prime}}}\left( \rho \right)=\frac{{\text{d}}}{{{\text{d}}\rho }}{\text{U}} \equiv \alpha \left( { - \frac{{2x}}{{{y^3}}}{{\text{U}}_1}+\frac{{2{x^2} - {y^2}}}{{{y^3}}}{{\text{U}}_2}} \right)$$

(B13)

To simplify further differentiation, the identity ${y^2}={x^2} - 1$ is applied to (B13) so that the numerator depends only on x, and the denominator only on y, giving

$${{\text{U}^{\prime}}}\left( \rho \right)=\alpha \left( { - \frac{{2x}}{{{y^3}}}{{\text{U}}_1}+\frac{{{x^2}+1}}{{{y^3}}}{{\text{U}}_2}} \right)$$

(B14)

Evaluating at ρ = ρ_e (where x = x_e, y = y_e), and applying condition (29b), results in

$$- 2{x_\text{e} }{{\text{U}}_1}+\left( {2x_{\text{e} }^{2} - y_{\text{e} }^{2}} \right){{\text{U}}_2}=0$$

(B15)

Solving the linear system formed by (B8) and (B15) yields

$${{\text{U}}_1}=\left( {x_{\text{e} }^{2}+1} \right){D_\text{e} }$$

(B16)

$${{\text{U}}_2}=2{x_\text{e} }{D_\text{e} }$$

(B17)

Next, applying operator (B12) to (B14) gives the second derivative

$${{\text{U}^{\prime\prime}}}\left( \rho \right)={\alpha ^2}\left( {\frac{{4{x^2}+2}}{{{y^4}}}{{\text{U}}_1} - \frac{{{x^3}+5x}}{{{y^4}}}{{\text{U}}_2}} \right)$$

(B18)

At equilibrium ρ = ρ_e, using (B16), (B17), (B18) with condition (29c) produces

$${{\text{U}^{\prime\prime}}}\left( {{\rho _\text{e} }} \right)=2{\alpha ^2}{D_\text{e} }$$

(B19)

$$\alpha ={\text{\varvec{\uppi}}}c{\omega _{\text{e}}}\sqrt {\frac{{2\mu }}{{{D_{\text{e}}}}}}$$

(B20)

Proceeding similarly, the third derivative at equilibrium is obtained as

$${{\text{U}^{\prime\prime\prime}}}\left( {{\rho _\text{e} }} \right)= - 6{\alpha ^3}{D_\text{e} }\frac{{{x_\text{e} }}}{{{y_\text{e} }}} \equiv - 6{\alpha ^3}{D_\text{e} }\coth \alpha \left( {{\rho _\text{e} } - {\rho _0}} \right)$$

(B21)

Combining (B19), (B21), and condition (29d), and solving gives

$${\rho _0}={\rho _\text{e} } - \frac{1}{\alpha }\ln \sqrt {\left| {\frac{{1+\tfrac{{{\alpha _\text{e} }{\omega _\text{e} }}}{{6{\text{B}}_{{\text{e}}}^{{\text{2}}}}}+\alpha {\rho _\text{e} }}}{{1+\tfrac{{{\alpha _\text{e} }{\omega _\text{e} }}}{{6{\text{B}}_{{\text{e}}}^{{\text{2}}}}} - \alpha {\rho _\text{e} }}}} \right|}$$

(B22)

where |x| denotes the absolute value of x.

The Varshni conditions determine expressions for U₁, U₂, _α, and ρ₀, but do not yield a direct formula for U₀. To obtain U₀, the constraint U(ρ_e) = 0 is imposed on the potential, giving U₀ = D_e. This choice shifts the potential minimum to zero without altering the physical observables of the system⁴⁸. The motivation for this constraint is: (a) It standardizes the reference energy for comparison with other diatomic oscillator models. (b) It preserves all thermodynamic and spectroscopic properties of the modeled system. This convention ensures that the equilibrium minimum of the EPT oscillator is fixed at zero potential while maintaining consistency with established results in the literature.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

Reprints and permissions

About this article

Cite this article

Eyube, E.S., Kamaludeen, M., Vijinti, F.C. et al. Energy and thermal magnetization of diatomic molecules under the enhanced Pöschl-Teller potential. Sci Rep (2026). https://doi.org/10.1038/s41598-026-49748-w

Download citation

Received: 17 November 2025
Accepted: 16 April 2026
Published: 25 April 2026
DOI: https://doi.org/10.1038/s41598-026-49748-w