Thermodynamic Properties of Diatomic Molecules: A Jost Function Approach
DOI:
https://doi.org/10.15407/ujpe71.8.641Keywords:
Jost function, diatomic molecule, partition function, internal energy, free energy, specific heat, entropyAbstract
A differential equation approach was employed to derive the necessary expression for the regular solution. By utilizing the interplay between the regular solution, the irregular solution, and the Jost function, the Jost function for the Manning–Rosen potential was constructed in its most reduced form. This Jost function was then applied to investigate the bound state energies and thermodynamic properties of both homonuclear and heteronuclear diatomic molecules.
References
1. A.M. Desai, N. Mesquita, V. Fernandes. A new modified Morse potential energy function for diatomic molecules. Phys. Scr. 95, 085401 (2020).
https://doi.org/10.1088/1402-4896/ab9bdc
2. O. Klein. Über die bandenspektren von helium und wasserstoff. Z. Physik 76, 226 (1932).
https://doi.org/10.1007/BF01341814
3. R. Rydberg. Graphische darstellung einiger bandenspektroskopischer ergebnisse. Z. Physik 73, 376 (1931).
https://doi.org/10.1007/BF01341146
4. R. Rydberg. Über einige potentialkurven des quecksilberhydrids. Z. Physik 80, 514 (1933).
https://doi.org/10.1007/BF02057312
5. E.J. Rosenbaum. Potential energy curve of the excited state of LiH. J. Chem. Phys. 6, 16 (1938).
https://doi.org/10.1063/1.1750117
6. G.M. Almy, A.C. Beiler. Potential energy curve of the excited state of KH. Phys. Rev. 61, 476 (1942).
https://doi.org/10.1103/PhysRev.61.476
7. F.H. Crawford, T. Jorgenson. The dissociation of hydrogen molecules by electron impact. Phys. Rev. 49, 745 (1936).
8. S.K. Chakravarty. Quantization under two centres of forces. Part I. The hydrogen molecular ion. Phil. Mag. 28, 423 (1939).
https://doi.org/10.1080/14786443908521198
9. H.M. James, A.S. Coolidge. The ground state of the hydrogen molecule. J. Chem. Phys. 1, 825 (1933).
https://doi.org/10.1063/1.1749252
10. E. Teller. Zur Theorie der Molekülbildung. Z. Physik 61, 458 (1930).
https://doi.org/10.1007/BF01330302
11. P.M. Morse. Diatomic molecules according to the wave mechanics. II. Vibrational Levels. Phys. Rev. 34, 57 (1929).
https://doi.org/10.1103/PhysRev.34.57
12. A.A. Frost, B. Musulin. Semiempirical potential energy functions. I. The H₂ and H₂⁺ diatomic molecules. J. Chem. Phys. 22, 1017 (1954).
https://doi.org/10.1063/1.1740254
13. E.R. Lippincott, R. Schroeder. General relation between potential energy and internuclear distance for diatomic and polyatomic molecules. J. Chem. Phys. 23, 1131 (1955).
https://doi.org/10.1063/1.1742201
14. N. Rosen, P.M. Morse. On the vibrations of polyatomic molecules. Phys. Rev. 42, 210 (1932).
https://doi.org/10.1103/PhysRev.42.210
15. V.S. Allured, C.M. Kelly, C.R. Landis. SHAPES empirical force field: New treatment of angular potentials and its application to square-planar transition-metal complexes. J. Am. Chem. Soc. 113, 1 (1991).
https://doi.org/10.1021/ja00001a001
16. A.K. Rappe, C.J. Casewit, K.S. Colwell, W.A. Goddard, W.M. Skiff. UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J. Am. Chem. Soc. 114, 10024 (1992).
https://doi.org/10.1021/ja00051a040
17. S. Barlow, A.A. Rohl, S. Shi, C.M. Freeman, D. O'Hare. Molecular mechanics study of oligomeric models for poly(ferrocenylsilanes) using the extensible systematic force field (ESFF). J. Am. Chem. Soc. 118, 7875 (1996).
https://doi.org/10.1021/ja953680s
18. 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, 014101 (2012).
https://doi.org/10.1063/1.4731340
19. P.Q. Wang, L.H. Zhang, C.S. Jia, J.Y. Liu. Equivalence of the three empirical potential energy models for diatomic molecules. J. Mol. Spectrosc. 274, 5 (2012).
https://doi.org/10.1016/j.jms.2012.03.005
20. P.Q. Wang, J.Y. Liu, L.H. Zhang, S.Y. Cao, C.S. Jia. Improved expressions for the Schiöberg potential energy models for diatomic molecules. J. Mol. Spectrosc. 278, 23 (2012).
https://doi.org/10.1016/j.jms.2012.07.001
21. C. Berkdemir, A. Berkdemir, R. Sever. Polynomial solutions of the Schrödinger equation for the generalized Woods-Saxon potential. Phys. Rev. C 72, 027001 (2005).
https://doi.org/10.1103/PhysRevC.72.027001
22. G.D. Zhang, J.Y. Liu, L.H. Zhang, W. Zhou, C.S. Jia. Modified Rosen-Morse potential-energy model for diatomic molecule. Phys. Rev. A 86, 062510 (2012).
https://doi.org/10.1103/PhysRevA.86.062510
23. N. Zettili. Quantum Mechanics: Concepts and Applications (Wiley, India, 2016).
24. R. Shankar. Principles of Quantum Mechanics (Plenum Press, 2011).
25. A.I. Ahmadov, Sh.M. Nagiyev, C. Aydin, V.A. Tarverdiyeva, M.Sh. Orujova, S.V. Badalov. Bound state solutions of Dirac equation: Spin and pseudo-spin symmetry in the presence of the combined Manning-Rosen and Yukawa tensor potentials. Eur. Phys. J. Plus 137, 1075 (2022).
https://doi.org/10.1140/epjp/s13360-022-03255-9
26. E.P. Inyang, E.S. William, J.A. Obu. Eigensolutions of the N-dimensional Schrödinger equation interacting with Varshni-Hulthén potential model. Rev. Mex. Fis. 67(2), 193 (2021).
https://doi.org/10.31349/RevMexFis.67.193
27. I.B. Okon, O.O. Popoola, E. Omugbe, A.D. Anita, C.N. Isonguyo, E.E. Ituen. Thermodynamic properties and bound state solutions of Schrödinger equation with Möbius square plus screened-Kratzer potential using Nikiforov-Uvarov method. Comput. Theor. Chem. 1196, 113132 (2021).
https://doi.org/10.1016/j.comptc.2020.113132
28. A.J. Sous. The asymptotic iteration method for the eigenenergies of a novel hyperbolic single wave potential. J. Appl. Math. Phys. 3, 1406 (2015).
https://doi.org/10.4236/jamp.2015.311168
29. S.M. Ikhdair, B.J. Falaye. Approximate analytical solutions to relativistic and nonrelativistic Pöschl-Teller potential with its thermodynamic properties. Chem. Phys. 421, 84 (2013).
https://doi.org/10.1016/j.chemphys.2013.05.021
30. S.Kr. Nayek. Bound state solutions of diatomic molecules with screened modified Kratzer potential by asymptotic iteration method. Eur. Phys. J. Plus 137, 1205 (2022).
https://doi.org/10.1140/epjp/s13360-022-03425-9
31. C.A. Onate, J.O. Ojomubah. Eigensolutions of the Schrödinger equation with a class of Yukawa potentials via supersymmetric approach. J. Theor. Appl. Phys. 10, 21 (2016).
https://doi.org/10.1007/s40094-015-0196-2
32. C.A. Onate, M.C. Onyeaju, A.N. Ikot, J.O. Ojomubah. Analytical solutions of the Klein-Gordon equation with a combined potential. Chin. J. Phys. 54, 820 (2016).
https://doi.org/10.1016/j.cjph.2016.08.007
33. H. Hassanabadi, B.H. Yazarloo, M. Mahmoudieh, S. Zarrinkamar. Dirac equation under the Deng-Fan potential and the Hulthén potential as a tensor interaction via SUSYQM. Eur. Phys. J. Plus 128, 111 (2013).
https://doi.org/10.1140/epjp/i2013-13111-4
34. E. Maghsoodi, H. Hassanabadi, S. Zarrinkamar, H. Rahimov. Relativistic symmetries of the Dirac equation under the nuclear Woods-Saxon potential. Phys. Scr. 85, 055007 (2012).
https://doi.org/10.1088/0031-8949/85/05/055007
35. K.J. Oyewumi, B.J. Falaye, C.A. Onate, O.J. Oluwadare, W.A. Yahya. Thermodynamic properties and the approximate solutions of the Schrödinger equation with the shifted Deng-Fan potential model. Mol. Phys. 112, 127 (2014).
https://doi.org/10.1080/00268976.2013.804960
36. M. Abu-Shady, E.M. Khokha, T.A. Abdel-Karim. The generalized fractional NU method for the diatomic molecules in the Deng-Fan model. Eur. Phys. J. D 76(9), 159 (2022).
https://doi.org/10.1140/epjd/s10053-022-00480-w
37. M. Abu-Shady, M.M.A. Ahmed, N.H. Gerish. Generalized fractional of the extended Nikiforov-Uvarov method for heavy tetraquark masses spectra. Mod. Phys. Lett. A 38(04), 2350028 (2023).
https://doi.org/10.1142/S0217732323500281
38. M. Abu-Shady, N.H. Gerish. The spectra masses for heavy pentaquark using generalized fractional of the extended Nikiforov-Uvarov method. Rev. Mex. Fis. 70, 030801 (2024).
https://doi.org/10.31349/RevMexFis.70.030801
39. T. Imbo, A. Pagnamenta, U. Sukhatme. Energy eigenstates of spherically symmetric potentials using the shifted expansion. Phys. Rev. D 29(8), 1669 (1984).
https://doi.org/10.1103/PhysRevD.29.1669
40. H. Scherrer, H. Risken, T. Leiber. Eigenvalues of the Schrödinger equation with rational potentials. Phys. Rev. A 38(8), 3949 (1988).
https://doi.org/10.1103/PhysRevA.38.3949
41. E. Papp, C. Micu. The derivation of 1/N equivalent potentials for the radial Schrödinger equation in N space dimensions. Can. J. Phys. 74, 102 (1996).
https://doi.org/10.1139/p96-016
42. C. Berkdemir, A. Berkdemir, J. Han. Bound state solutions of the Schrödinger equation for modified Kratzer's molecular potential. Chem. Phys. Lett. 417, 326 (2006).
https://doi.org/10.1016/j.cplett.2005.10.039
43. C.O. Edet, U.S. Okorie, A.T. Ngiangia, A.N. Ikot. Bound state solutions of the Schrödinger equation for the modified Kratzer potential plus screened Coulomb potential. Indian J. Phys. 94, 425 (2020).
https://doi.org/10.1007/s12648-019-01477-9
44. E.E. Ibekwe, A.T. Ngiangia, U.S. Okorie, A.N. Ikot, H.Y. Abdullah. Bound state solution of radial Schrödinger equation for the quark-antiquark interaction potential. Iran. J. Sci. Technol. Trans. A 44, 1191 (2020).
https://doi.org/10.1007/s40995-020-00913-4
45. S.S. Alves, F.D.S. Azevedo, C. Filgueiras, E.O. Silva. Exact and approximate bound state solutions of the Schrödinger equation with a class of Kratzer-type potentials in the global monopole spacetime. Chin. J. Phys. 88, 609 (2024).
https://doi.org/10.1016/j.cjph.2023.10.012
46. B.I. Ita, H. Louis, E.I. Ubana, P.E. Ekuri, C.U. Leonard, N.I. Nzeata. Evaluation of the bound state energies of some diatomic molecules from the approximate solutions of the Schrödinger equation with Eckart plus inversely quadratic Yukawa potential. J. Mol. Model. 26, 1 (2020).
https://doi.org/10.1007/s00894-020-04593-0
47. U.S. Okorie, E.E. Ibekwe, A.N. Ikot, M.C. Onyeaju, E.O. Chukwuocha. Thermodynamic properties of the modified Yukawa potential. J. Korean Phys. Soc. 73, 1211 (2018).
https://doi.org/10.3938/jkps.73.1211
48. R. Jost. Über die falschen Nullstellen der Eigenwerte der S-matrix. Helv. Phys. Acta 20, 256 (1947).
49. D. Saha, B. Khirali, B. Swain, J. Bhoi. Jost states for the Deng-Fan potential. Phys. Scr. 98(1), 015303 (2022).
https://doi.org/10.1088/1402-4896/aca1e6
50. A.K. Behera, U. Laha, P. Sahoo, J. Bhoi. Hulthén half-off-shell t matrix-application to n-p and n-d systems. J. Korean Phys. Soc. 76, 782 (2020).
https://doi.org/10.3938/jkps.76.782
51. P. Sahoo, U. Laha, B. Swain. Off-shell scattering by an approximated additive interaction. Pramana J. Phys. 97(2), 62 (2023).
https://doi.org/10.1007/s12043-023-02519-y
52. J. Bhoi, A.K. Behera, U. Laha. Off-shell Jost function for the Hulthén potential in all partial waves. J. Math. Phys. 60, 083502 (2019).
https://doi.org/10.1063/1.5093115
53. U. Laha, J. Bhoi. On- and off-shell Jost functions and their integral representations. Pramana 86, 947 (2016).
https://doi.org/10.1007/s12043-015-1130-5
54. R.G. Newton. Scattering Theory of Waves and Particles. 2nd ed. (Springer-Verlag, 1982).
https://doi.org/10.1007/978-3-642-88128-2
55. S.A. Rakityansky. Jost Functions in Quantum Mechanics: A Unified Approach to Scattering, Bound, and Resonant State Problems (Springer Nature Switzerland, 2022).
https://doi.org/10.1007/978-3-031-07761-6
56. U. Laha, J. Bhoi. Off-shell Jost solutions for Coulomb and Coulomb-like interactions in all partial waves. J. Math. Phys. 54, 013514 (2013).
https://doi.org/10.1063/1.4776659
57. M.F. Manning, R. Rosen. A potential function for the vibrations of diatomic molecules. Phys. Rev. 44, 953 (1933).
58. B. Khirali, A.K. Behera, J. Bhoi, U. Laha. Regular and Jost states for the S-wave Manning-Rosen potential. J. Phys. G: Nucl. Part. Phys. 46, 115104 (2019).
https://doi.org/10.1088/1361-6471/ab4118
59. P.R. Kumar. New wine in old bottle: An improved perspective on the Greene-Aldrich approximation for the generalized Pöschl-Teller potential. Russ. Phys. J. 67, 1204 (2024).
https://doi.org/10.1007/s11182-024-03234-w
60. A.W. Babister. Transcendental Functions Satisfying Non-Homogeneous Linear Differential Equations (MacMillan, 1967).
61. L.J. Slater. Confluent Hypergeometric Functions (Cambridge University Press, 1960).
https://doi.org/10.2307/2003114
62. W.A. Yahya, K.J. Oyewumi. Thermodynamic properties and approximate solutions of the ℓ-state Pöschl-Teller-type potential. J. Assoc. Arab Univ. Basic Appl. Sci. 21, 53 (2016).
https://doi.org/10.1016/j.jaubas.2015.04.001
63. C.S. Jia, C.W. Wang, L.H. Zhang, X.L. Peng, R. Zeng, X.T. You. Partition function of improved Tietz oscillators. Chem. Phys. Lett. 676, 150 (2017).
https://doi.org/10.1016/j.cplett.2017.03.068
64. P.M. Morse, H. Feshbach. Methods of Theoretical Physics (McGraw-Hill, 1953).
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