Computation of vibrational partition function: a comparative analysis between Poisson summation and classical limit

Authors

  • C. A. Onate
    Physics Department, Bowen University, Iwo, Nigeria
  • I. B. Okon
    Department of Physics, University of Uyo, Uyo, Nigeria
  • E. S. Eyube
    Department of Physics, Faculty of Physical Sciences, Modibbo Adama University, Yola, Nigeria
  • E. Omugbe
    Department of Physics, University of Agriculture and Environmental Sciences, Umuagwo, Nigeria
  • A. D. Ahmed
    Department of Physics, Faculty of Physical Sciences, Modibbo Adama University, Yola, Nigeria

Keywords:

Thermodynamic properties, Entropy, Enthalpy, Gibbs free energy

Abstract

The computation of partition function (Z) is the bedrock of the study of statistical mechanics as it plays a significant role in the thermodynamic properties (TP) where microscopic properties are connected to macroscopic properties. Several studies have reported TP via the Z using one of the classical limit approach or Poisson summation formula. No study however justifies the agreement or discrepancy between the methods. This study therefore, investigates the two methods for theoretical determination of the vibrational partition function by considering the energy levels of Tietz molecular potential. The first approach employs Poisson summation method with a defined energy levels of the Tietz molecular potential while the second approach adopts the utilization of the classical limit approach with the same energy levels of the Tietz molecular potential. By comparing the results of the two approaches, our result reveals discrepancy between the analytic equations for Z. However, the numerical results obtained for the thermal properties of NaBr and CuCl molecules showed a perfect agreement between the two approaches and the experimental data with the results from classical limit approach closer to the experimental data. This study therefore, provides guidelines for choosing the appropriate approach based on the characteristics of the system under study for theoretical research.

Dimensions

[1] L.-H. Zhang, X.-P. Li and C.-S. Jia, “Approximate solutions of the Schrodinger equation with the generalized Morse potential model including the centrifugal term”, International Journal of Quantum Chemistry 111 (2011) 1870. https://doi.org/10.1002/qua.22477.

[2] C. A. Onate, A. N. Ikot, M. C. Onyeaju, O. Ebomwonyi and J. O. A. Idiodi, “Effect of dissociation energy on Shannon and Renyi entropies”, Karbala International Journal of Modern Science 4 (2018) 134. https://doi.org/10.1016/j.kijoms.2017.12.004.

[3] K. J. Oyewumi, B. J. Falaye, C. A. Onate, O. J. Oluwadare and W. A. Yahya, “Thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng?Fan potential model”, Molecular Physics 112 (2014) 127. https://doi.org/10.1080/00268976.2013.804960.

[4] O. J. Oluwadare and K. J. Oyewumi, “Energy spectra and the expectation values of diatomic molecules confined by the shifted Deng-Fan potential”, The European Physical Journal Plus 133 (2018) 422. https://doi.org/10.48550/arXiv.1806.00103.

[5] C. A. Onate, G. O. Egharevba and D. T. Bankole, “Eigensolution to Morse potential for Scandium and Nitrogen monoiodides”, Journal of the Nigerian Society of Physical Sciences 3 (2021) 282. https://doi.org/10.46481/jnsps.2021.407.

[6] O. J. Oluwadare and K. J. Oyewumi, “Scattering state solutions of the Duffin-Kemmer-Petiau equation with the Varshni potential model”, The European Physical Journal A 53 (2017) 29. https://doi.org/10.1140/epja/i2017-12218-5.

[7] W. A. Yahya and K. J. Oyewumi, “Thermodynamic properties and approximate solutions of the ℓ-state Poschl?Teller-type potential”, Journal of the Association of Arab Universities for Basic and Applied Sciences 21 (2016) 53. https://doi.org/10.1016/j.jaubas.2015.04.001.

[8] M. Servatkhah, R. Khordad and A. Ghanbari, “Accurate prediction of thermodynamic functions of H2 and LiH using theoretical calculations”, International Journal of Thermodynamics 41 (2020) 37. https://doi.org/10.1007/s10765-020-2615-0.

[9] N. M. Yunus, M. I. A. Mutalib, Z. Man, M. A. Bustam and T. Murugesan, “Solubility of CO2 in pyridinium based ionic liquids”, Chemical Engineering Journal 189 (2012) 94. https://doi.org/10.1016/j.cej.2012.02.033.

[10] S. A. Moses, J. P. Covey, M. T. Miecnikowski, B. Yan, B. Gadway, J. Ye and D. S. Jin, “Creation of a low-entropy quantum gas of polar molecules in an optical lattice”, Science 350 (2015) 659. https://doi.org/10.1126/science.aac6400.

[11] P. Ammendola, F. Raganati and R. Chirone, “CO2 Adsorption on a fine activated carbon in a sound assisted fluidized bed: thermodynamics and kinetics”, Chemical Engineering Journal 322 (2017) 302. https://doi.org/10.1016/j.cej.2017.04.037.

[12] D. E. Otten, P. R. Shaffer, P. L. Geissler and R. J. Saykally, “Elucidating the mechanism of selective ion adsorption to the liquid water surface”, Proceedings of the National Academy of Science USA 109 (2012) 701. https://doi.org/10.1073/pnas.1116169109.

[13] I. Yahiatne, S. Hennig and T. Huser, “Optical fluctuation microscopy based on calculating local entropy values”, Chemical Physics Letters 587 (2013) 1. https://doi.org/10.1016/j.cplett.2013.08.102.

[14] N. P. Stadie, M. Murialdo, C. C. Ahn and B. Fultz, “Anomalous isosteric enthalpy of adsorption of methane on zeolite-templated carbon supporting information”, Journal of American Chemical Society 135 (2013) 990. https://doi.org/10.1021/ja311415m.

[15] W. Song, N. Martsinovich, W. M. Heckl and M. Lackinger, “Thermodynamics of halogen bonded monolayer self-assembly at the liquid-solid interface”, Chemical Communications 50 (2014) 13465. https://doi.org/10.1039/c4cc06251e.

[16] T. Zhang, G. S. Ellis, S. C. Ruppel, K. Milliken and R. Yang, “Effect of organic-matter type and thermal maturity on methane adsorption in shale-gas systems”, Organic Geochemistry 47 (2012) 120. https://doi.org/10.1016/j.orggeochem.2012.03.012.

[17] S. Dastidar, C. J. Hawley, A. D. Dillon, A. D. Gutierrez-Perez, J. E. Spanier and A. T. Fafarman, “Quantitative phase-change thermodynamics and metastability of perovskite-phase cesium lead iodide”, Journal of Physics and Chemical Letters 8 (2017) 1278. https://doi.org/10.1021/acs.jpclett.7b00134.

[18] A. N. Ikot, U. S. Okorie, R. Sever and G. J. Rampho, “Eigensolution, expectation values and thermodynamic properties of the screened Kratzer potential”, The European Physical Journal Plus 134 (2019) 386. https://doi.org/10.1140/epjp/i2019-12783-x.

[19] A. N. Ikot, C. O. Edet, P. O. Amadi, U. S. Okorie, G. J. Rampho, and H. Y. Abdullah, “Thermodynamic properties of Aharanov Bohm (AB) and magnetic fields with screened Kratzer potential”, The European Physical Journal D 74 (2020) 159. https://doi.org/10.1140/epjd/e2020-10084-9.

[20] A. N. Ikot, B. C. Lutfuoglu, M. I. Ngwueke, M. E. Udoh, S. Zare and H. Hassanabadi, “Klein-Gordon equation particles in exponential-type molecule potentials and its thermodynamic properties in D-dimensions”, European Physical Journal Plus 131 (2016) 1. https://doi.org/10.1140/epjp/i2016-16419-5.

[21] I. B. Okon, E. Omugbe, A. D. Antia, C. A. Onate, L. E. Akpabio and O. E. Osafile, “Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential”, Scientific Reports 11 (2021) 892. https://doi.org/10.1038/s41598-020-77756-x.

[22] I. B. Okon, C. A. Onate, E. Omugbe, U. S. Okorie, C. O. Edet, A. D. Antia, J. P. Araujo, C. N. Isonguyo, M. C. Onyeaju, E. S. William, R. Horchani and A. N. Ikot, “Aharonov?Bohm (AB) flux and thermomagnetic properties of Hellmann plus screened Kratzer potential as applied to diatomic molecules using Nikiforov?Uvarov-Functional-Analysis (NUFA) method”, Molecular Physics 120 (2022) e2046295. https://doi.org/10.1080/00268976.2022.2046295.

[23] I. B. Okon, O. O. Popoola, E. Omugbe, A. D. Antia, C. N. Isonguyo and E. E. Ituen, “Thermodynamic properties and bound state solutions of Schrodinger equation with Mobius square plus screened-Kratzer potential using Nikiforov-Uvarov method”, Computational and Theoretical Chemistry 1196 (2021) 113132. https://doi.org/10.1016/j.comptc.2020.113132.

[24] E. S. Eyube, B. M. Bitrus and Y. Y. Jabil, “Thermodynamic relations and ro-vibrational energy levels of the improved Pöschl?Teller oscillator for diatomic molecules”, Journal of Physics B: Atomic, Molecular and Optical Physics 54 (2021) 155102. https://doi.org/10.1088/1361-6455/ac00c5.

[25] E. S. Eyube, P. P. Notani, Y. Dlama, E. Omugbe, C. A. Onate, I. B. Okon, G. G. Nyam, Y. Y. Jabil and M. M. Izam, “Isobaric molar heat capacity model for the improved Tietz potential”, International Journal of Quantum Chemistry 123 (2023) e27040. https://doi.org/10.1002/qua.27040.

[26] Q.-C. Ding, C.-S. Jia, C.-W. Wang, X.-L. Peng, J.-Y. Liu, L.-H. Zhang, R. Jiang, S.-Y. Zhu, H. Yuan and H.-X. Tang, “Unified non-fitting formulation representation of thermodynamic properties for diatomic substances”, Journal of Molecular Liquids 371 (2023) 121088. https://doi.org/10.1016/j.molliq.2023.121088.

[27] Q.-C. Ding, C.-S. Jia, J.-Z. Liu, J. Li, R.-F. Du, J.-Y. Liu, X.-L. Peng, C.-W. Wang and H.-X. Tang, “Prediction of thermodynamic properties for sulfur dimer”, Chemical Physics Letters 803 (2022) 139844. https://doi.org/10.1016/j.cplett.2022.139844.

[28] G.-H. Liu, Q.-C. Ding, C.-W. Wang and C.-S. Jia, “Unified non-fitting explicit formulation of thermodynamic properties for five compounds”, Journal of Molecular Structure 1294 (2023) 136543. https://doi.org/10.1016/j.molstruc.2023.136543.

[29] D.-C. Liang, R. Zeng, C.-W. Wang, Q.-C. Ding, L.-S. Wei, X.-L. Peng, J.-Y. Liu, J. Yu and C.-S. Jia, “Prediction of thermodynamic properties for sulfur dioxide”, Journal of Molecular Liquids 352 (2022) 118722. https://doi.org/10.1016/j.molliq.2022.118722.

[30] C.-W. Wang, X.-L. Peng, J.-Y. Liu, R. Jiang, X.-P. Li, Y.-S. Liu, S.-Y. Liu, L.-S. Wei, L.-H. Zhang and C.-S. Jia, “A novel formulation representation of the equilibrium constant for water gas shift reaction”, International Journal of Hydrogen Energy 47 (2022) 27821. https://doi.org/10.1016/j.ijhydene.2022.06.105.

[31] C.-W. Wang, J. Li, L.-H. Zhang, Q.-C. Ding, G. H. Liu, G. Li, R. Jiang, X.-L. Peng, L.-S. Wei, H.-M. Tang, J.-Y. Liu and C.-S. Jia, “Non-fitting functional representation for the equilibrium constant subject to reaction between H2 S and CO2 ”, Fuel 362 (2024) 130916. https://doi.org/10.1016/j.fuel.2024.130916.

[32] A. Ghanbari and R. Khordad, “Theoretical prediction of thermodynamic properties of N2 and CO using pseudoharmonic and Mie-type potentials”, Chemical Physics 534 (2020) 110732. https://doi.org/10.1016/j.chemphys.2020.110732.

[33] C. S. Jia, Y. F. Diao, X. J. Liu, P. Q. Wang, J. Y. Liu, G. D. Zhang, “Equivalence of the Wei potential model and Tietz potential model for diatomic molecules”, J. Chem. Phys. 137 (2012) 014101. http://dx.doi.org/10.1063/1.4731340.

[34] R. Khordad and H. R. R. Sedehi, “Thermodynamic properties of a double ring-shaped quantum dot at low and high temperatures”, Journal of Low Temperature Physics 190 (2018) 200. https://doi.org/10.1007/s10909-017-1831-x.

[35] R. Khordad and A. Ghanbari, “Theoretical prediction of thermodynamic functions of TiC: Morse ring-shaped potential”, Journal of Low Temperature Physics 199 (2020) 1198. https://doi.org/10.1007/s10909-020-02368-8.

[36] R. Khordad and A. Ghanbari, “Analytical calculations of thermodynamic functions of lithium dimer using modified Tietz and Badawi-Bessis-Bessis potentials”, Computational and Theoretical Chemistry 1155 (2019) 1. https://doi.org/10.1016/j.comptc.2019.03.019.

[37] E. S. Eyube, C. A. Onate, E. Omugbe and C. M. Nwabueze, “Theoretical prediction of Gibbs free energy and specific heat capacity of gaseous molecules”, Chemical Physics 560 (2022) 111572. https://doi.org/10.1016/j.chemphys.2022.111572.

[38] G.-H. Liu, Q.-C. Ding, C.-W. Wang, C.-S. Jia, “Unified explicit formulations of thermodynamic properties for the gas NO2 , and gaseous BF2 and AlCl2 radicals”, Chemical Physics Letters 830 (2023) 140788. https://doi.org/10.1016/j.cplett.2023.140788.

[39] C.-S. Jia, C.-W. Wang, L.-H. Zhang, X.-L. Peng, H.-M. Tang, R. Zeng, “Enthalpy of gaseous phosphorus dimer”, Chemical Engineering Science 183 (2018) 26. https://doi.org/10.1016/j.ces.2018.03.009.

[40] C.-S. Jia, J. Li, Y.-S. Liu, X.-L. Peng, X. Jia, L.-H. Zhang, R. Jiang, X.-P. Li, J.-Y. Liu, Y.-L. Zhao, “Predictions of thermodynamic properties for hydrogen sulfide”, Journal of Molecular Liquids 315 (2020) 113751. https://doi.org/10.1016/j.molliq.2020.113751.

[41] C.-S. Jia, C.-W. Wang, L.-H. Zhang, X.-L. Peng, R. Zeng and X.-T. You, “Partition function of improved Tietz oscillators”, Chemical Physics Letters 676 (2017) 153. https://doi.org/10.1016/j.cplett.2017.03.068.

[42] C.-S. Jia, L.-H. Zhang, X.-L. Peng, J.-X. Luo, Y.-L. Zhao, J.-Y. Liu, J.-J. Guo and L.-D. Tang, “Prediction of entropy and Gibbs free energy for nitrogen”, Chemical Engineering Science 202 (2019) 70. https://doi.org/10.1016/j.ces.2019.03.033.

[43] C.-S. Jia, R. Zeng, X.-L. Peng, L.-H. Zhang, Y.-L. Zhao, “Entropy of gaseous phosphorus dimer”, Chemical Engineering Science 190 (2018) 1. https://doi.org/10.1016/j.ces.2018.06.009.

[44] C.-S. Jia, C.-W. Wang, L.-H. Zhang, X.-L. Peng, H.-M. Tang, J.-Y. Liu, Y. Xiong, R. Zeng, “Predictions of entropy for diatomic molecules and gaseous substances”, Chemical Physics Letters 692 (2018) 57. https://doi.org/10.1016/j.cplett.2017.12.013.

[45] S.-H. Dong, M. Lozada-Cassou, J. Yu, F. Jimenez-Ángeles, A. L. Rivera, “Hidden symmetries and thermodynamic properties for a harmonic oscillator plus an inverse square potential”, International Journal of Quantum Chemistry 107 (2007) 366. https://doi.org/10.1002/qua.21103.

[46] S.-H. Dong, R. Lemus, A. Frank, “Ladder operators for the Morse potential”, International Journal of Quantum Chemistry 86 (2002) 433. https://doi.org/10.1002/qua.10038.

[47] S.-H. Dong, “The SU(2) realization for the Morse potential and its coherent states”, Canadian Journal of Physics 80 (2002) 129. https://doi.org/10.1139/p01-130.

[48] S.-H. Dong and M. Cruz-Irisson, “Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties”, Journal of Mathematical Chemistry 50 (2012) 881. https://doi.org/10.1007/s10910-011-9931-3.

[49] M. L. Strekalov, “An accurate closed-form expression for the partition function of Morse oscillators”, Chemical Physics Letters 439 (2007) 209. https://doi.org/10.1016/j.cplett.2007.03.052.

Published

2026-02-01

How to Cite

Computation of vibrational partition function: a comparative analysis between Poisson summation and classical limit. (2026). Journal of the Nigerian Society of Physical Sciences, 8(1), 2834. https://doi.org/10.46481/jnsps.2026.2834

Issue

Volume 8, Issue 1, February 2026

Section

Physics & Astronomy
Copyright & Licensing

Copyright (c) 2025 C.A. Onate, I.B. Okon, E.S. Eyube, E. Omugbe, A.D. Ahmed (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Computation of vibrational partition function: a comparative analysis between Poisson summation and classical limit. (2026). Journal of the Nigerian Society of Physical Sciences, 8(1), 2834. https://doi.org/10.46481/jnsps.2026.2834

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