A Theoretical Study on Vibrational Energies of Molecular Hydrogen and Its Isotopes Using a Semi-classical Approximation

Redi Kristian Pingak; Albert Zicko Johannes; Fidelis Nitti; Meksianis Zadrak Ndii

doi:10.22146/ijc.63294

A Theoretical Study on Vibrational Energies of Molecular Hydrogen and Its Isotopes Using a Semi-classical Approximation

https://doi.org/10.22146/ijc.63294

Redi Kristian Pingak^(1*), Albert Zicko Johannes⁽²⁾, Fidelis Nitti⁽³⁾, Meksianis Zadrak Ndii⁽⁴⁾

(1) Department of Physics, Universitas Nusa Cendana, Jl. Adisucipto Penfui, Kupang 85001, Nusa Tenggara Timur, Indonesia
(2) Department of Physics, Universitas Nusa Cendana, Jl. Adisucipto Penfui, Kupang 85001, Nusa Tenggara Timur, Indonesia
(3) Department of Chemistry, Universitas Nusa Cendana, Jl. Adisucipto Penfui, Kupang 85001, Nusa Tenggara Timur, Indonesia; School of Chemistry, University of Melbourne, Masson Road, Parkville, Victoria 3052, Australia
(4) Department of Mathematics, Universitas Nusa Cendana, Jl. Adisucipto Penfui, Kupang 85001, Nusa Tenggara Timur, Indonesia
(*) Corresponding Author

Abstract

This study aims to apply a semi-classical approach using some analytically solvable potential functions to accurately compute the first ten pure vibrational energies of molecular hydrogen (H₂) and its isotopes in their ground electronic states. This study also aims at comparing the accuracy of the potential functions within the framework of the semi-classical approximation. The performance of the approximation was investigated as a function of the molecular mass. In this approximation, the nuclei were assumed to move in a classical potential. The Bohr-Sommerfeld quantization rule was then applied to calculate the vibrational energies of the molecules numerically. The results indicated that the first vibrational transition frequencies (v₁_ß0) of all hydrogen isotopes were consistent with the experimental ones, with a minimum percentage error of 0.02% for ditritium (T₂) molecule using the Modified-Rosen-Morse potential. It was also demonstrated that, in general, the Rosen-Morse and the Modified-Rosen-Morse potential functions were better in terms of calculating the vibrational energies of the molecules than Morse potential. Interestingly, the Morse potential was found to be better than the Manning-Rosen potential. Finally, the semi-classical approximation was found to perform better for heavier isotopes for all potentials applied in this study.

Keywords

semi-classical approximation; classical potential functions; hydrogen isotopes; Bohr-Sommerfeld quantization

Full Text:

Full Text PDF

References

[1] Puchalski, M., Komasa, J., and Pachucki, K., 2017, Relativistic corrections for the ground electronic state of molecular hydrogen, Phys. Rev. A, 95 (5), 052506.

[2] Korobov, V.I., Hilico, L., and Karr, J.P., 2017, Fundamental transitions and ionization energies of the hydrogen molecular ions with few ppt uncertainty, Phys. Rev. Lett., 118 (23), 233001.

[3] Liu, J., Sprecher, D., Jungen, C., Ubachs, W., and Merkt, F., 2010, Determination of the ionization and dissociation energies of the deuterium molecule (D₂), J. Chem. Phys., 132 (15), 154301.

[4] Dickenson, G.D., Niu, M.L., Salumbides, E.J., Komasa, J., Eikema, K.S.E., Pachucki, K., and Ubachs, W., 2013, Fundamental vibration of molecular hydrogen, Phys. Rev. Lett., 110 (19), 193601.

[5] Wójtewicz, S., Gotti, R., Gatti, D., Lamperti, M., Laporta, P., Jóźwiak, H., Thibault, F., Wcisło, P., and Marangoni, M., 2020, Accurate deuterium spectroscopy and comparison with ab initio calculations, Phys. Rev. A, 101 (5), 052504.

[6] Lai, K.F., Hermann, V., Trivikram, T.M., Diouf, M., Schlösser, M., Ubachs, W., and Salumbides, E.J., 2020, Precision measurement of the fundamental vibrational frequencies of tritium-bearing hydrogen molecules: T₂, DT, HT, Phys. Chem. Chem. Phys., 22 (16), 8973–8987.

[7] Trivikram, T.M., Schlösser, M., Ubachs, W., and Salumbides, E.J., 2018, Relativistic and QED effects in the fundamental vibration of T₂, Phys. Rev. Lett., 120 (16), 163002.

[8] Lai, K.F., Czachorowski, P., Schlösser, M., Puchalski, M., Komasa, J., Pachucki, K., Ubachs, W., and Salumbides, E.J., 2019, Precision tests of nonadiabatic perturbation theory with measurements on the DT molecule, Phys. Rev. Res., 1 (3), 033124.

[9] Kalinin, K.V., 2011, Calculation of hydrogen molecule energy levels using the moment constant summability method with specialized weight, Atmos. Oceanic Opt., 24 (1), 17–21.

[10] Kurokawa, Y.I., Nakashima, H., and Nakatsuji, H., 2019, Solving the Schrödinger equation of hydrogen molecules with the free-complement variational theory: Essentially exact potential curves and vibrational levels of the ground and excited states of the Σ symmetry, Phys. Chem. Chem. Phys., 21 (12), 6327–6340.

[11] Kurokawa, Y.I., Nakashima, H., and Nakatsuji, H., 2020, Solving the Schrödinger equation of hydrogen molecules with the free-complement variational theory: Essentially exact potential curves and vibrational levels of the ground and excited states of the Π symmetry, Phys. Chem. Chem. Phys., 22 (24), 13489–13497.

[12] Desai, A.M., Mesquita, N., and Fernandes, V., 2020, A new modified Morse potential energy function for diatomic molecules, Phys. Scr., 95 (8), 085401.

[13] Pachucki, K., and Komasa, J., 2019, Non-relativistic energy levels of D₂, Phys. Chem. Chem. Phys., 21 (20), 10272–10276.

[14] Pachucki, K., and Komasa, J., 2018, Non-relativistic energy levels of HD, Phys. Chem. Chem. Phys., 20 (41), 26297–26302.

[15] Komasa, J., Puchalski, M., Czachorowski, P., Łach, G., and Pachucki, K., 2019, Rovibrational energy levels of the hydrogen molecule through nonadiabatic perturbation theory, Phys. Rev. A, 100 (3), 032519.

[16] Sholihun, S., Saito, M., Ohno, T., and Yamasaki, T., 2015, Density-functional-theory-based calculations of formation energy and concentration of the silicon monovacancy, Jpn. J. Appl. Phys., 54 (4), 041301.

[17] Sholihun, S., Kadarisman, H.P., and Nurwantoro, P., 2018, Density-functional-theory calculations of formation energy of the nitrogen-doped diamond, Indones. J. Chem., 18 (4), 749–754.

[18] Hastuti, D.P., Nurwantoro, P., and Sholihun, S., 2019, Stability study of germanene vacancies: The first-principles calculations, Mater. Today Commun., 19, 459-463.

[19] Amalia, W., Nurwantoro, P., and Sholihun, S., 2018, Density-functional-theory calculations of structural and electronic properties of vacancies in monolayer hexagonal boron nitride (h-BN), Comput. Condens. Matter, 18, e00354.

[20] Hutama, A.S., Huang, H., and Kurniawan, Y.S., 2019, Investigation of the chemical and optical properties of halogen-substituted N-methyl-4-piperidone curcumin analogs by density functional theory calculations, Spectrochim. Acta, Part A, 221, 117152.

[21] Al-Mushadani, O.K., and Needs, R.J., 2003, Free-energy calculations of intrinsic point defects in silicon, Phys. Rev. B, 68 (23), 235205.

[22] Sanati, M., and Estreicher, S.K., 2003, Defects in silicon: The role of vibrational entropy, Solid State Commun., 128 (5), 181–185.

[23] Hutama, A.S., Hijikata, Y., and Irle, S., 2017, Coupled cluster and density functional studies of atomic fluorine chemisorption on coronene as model systems for graphene fluorination, J. Phys. Chem. C, 121 (27), 14888–14898.

[24] Ceotto, M., Valleau, S., Tantardini, G.F., and Aspuru-Guzik, A., 2011, First principles semiclassical calculations of vibrational eigenfunctions, J. Chem. Phys., 134 (23), 234103.

[25] Vázquez, F.X., Talapatra, S., and Geva, E., 2011, Vibrational energy relaxation in liquid HCl and DCl via the linearized semiclassical method: Electrostriction versus quantum delocalization, J. Phys. Chem. A, 115 (35), 9775–9781.

[26] Walton, J.R., Rivera-Rivera, L.A., Lucchese, R.R., and Bevan, J.W., 2016, Morse, Lennard-Jones, and Kratzer potentials: A canonical perspective with applications, J. Phys. Chem. A, 120 (42), 8347–8359.

[27] Roy, A.K., 2013, Accurate ro-vibrational spectroscopy of diatomic molecules in a Morse oscillator potential, Results Phys., 3, 103–108.

[28] Pingak, R.K., and Johannes, A.Z., 2020, Penentuan tingkat-tingkat energi vibrasi molekul hidrogen pada keadaan elektronik dasar menggunakan potensial Morse, Wahana Fisika, 5 (1), 1–9.

[29] Morse, P.M., 1929, Diatomic molecules according to the wave mechanics II: Vibrational levels, Phys. Rev., 34 (1), 57–64.

[30] Rosen, N., and Morse, P.M., 1932, On the vibrations of polyatomic molecules, Phys. Rev., 42 (2), 210–217.

[31] Manning, M.F., and Rosen, N., 1933, A potential function for the vibrations of the diatomic molecules, Phys. Rev., 44, 953.

[32] Zhang, G.D., Liu, J.Y., Zhang, L.H., Zhou, W., and Jia, C.S., 2012, Modified Rosen-Morse potential-energy model for diatomic molecules, Phys. Rev. A, 86 (6), 062510.

[33] Deng, Z.H., and Fan, Y.P., 1957, A potential function of diatomic molecules, J. Shandong Univ., 1, 162–166.

[34] Liu, J.Y., Zhang, G.D., and Jia, C.S., 2013, Calculation of the interaction potential energy curve and vibrational levels of the a³Σ_u⁺ state of ⁷Li₂ molecule, Phys. Lett. A, 377 (21-22), 1444–1447.

[35] Wang, P.Q., Zhang, L.H., Jia, C.S., and Liu, J.Y., 2012, Equivalence of the three empirical potential energy models for diatomic molecules, J. Mol. Spectrosc., 274, 5-8.

[36] Koonin, S.E., and Meredith, D.C., 1998, Computational Physics: Fortran Version, 1^st Ed., CRC Press, Boca Raton, Florida, US.

[37] Jia, C.S., Diao, Y.F., Liu, X.J., Wang, P.Q., Liu, J.Y., and Zhang, G.D., 2012, Equivalence of the Wei potential model and Tietz potential model for diatomic molecules, J. Chem. Phys., 137 (1), 014101.

[38] Huber, K.P., and Herzberg, G., 1979, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules, 1^st Ed., Springer, US.

[39] Sprecher, D., Jungen, C., Ubachs, W., and Merkt, F., 2011, Towards measuring ionisation and dissociation energies of molecular hydrogen with sub-MHz accuracy, Faraday Discuss., 150, 51–70.

[40] Sprecher, D., Liu, J., Jungen, C., Ubachs, W., and Merkt, F., 2010, Communication: The ionization and dissociation energies of HD, J. Chem. Phys., 133 (11), 111102.

[41] Puchalski, M., Komasa, J., Spyszkiewicz, A., and Pachucki, K., 2019, Dissociation energy of molecular hydrogen isotopologues, Phys. Rev. A, 100 (2), 020503.

[42] Schwartz, C., and Le Roy, R.J., 1987, Nonadiabatic eigenvalues and adiabatic matrix elements for all isotopes of diatomic hydrogen, J. Mol. Spectrosc., 121 (2), 420–439.

[43] Royappa, A.T., Suri, V., and McDonough, J.R., 2006, Comparison of empirical closed-form functions for fitting diatomic interaction potentials of ground state first- and second-row diatomics, J. Mol. Struct., 787 (1-3), 209–215.

[44] Tang, H.M., Liang, G.C., Zhang, L.H., Zhao, F., and Jia, C.S., 2014, Diatomic molecule energies of the modified Rosen-Morse potential energy model, Can. J. Chem., 92 (4), 341–345.

[45] Niu, M.L., Salumbides, E.J., Dickenson, G.D., Eikema, K.S.E., and Ubachs, W., 2014, Precision spectroscopy of the X¹Σ_g⁺, v=0à1 (J=0”–“2) rovibrational splittings in H₂, HD and D₂, J. Mol. Spectrosc., 300, 44–54.

[46] Wolniewicz, L., 1983, The X¹Σ_g⁺ state vibration-rotation energies of the H₂, HD, and D₂ molecules, J. Chem. Phys., 78 (10), 6173-6181.

[47] Yanar, H., Aydoğdu, O., and Salti, M., 2016, Modelling of diatomic molecules, Mol. Phys., 114 (21), 3134–3142.

[48] Liu, J.Y., Hu, X.T., and Jia, C.S., 2014, Molecular energies of the improved Rosen-Morse potential energy model, Can. J. Chem., 92 (1), 40–44.

[49] Tan, M.S., He, S., and Jia, C.S., 2014, Molecular spinless energies of the improved Rosen-Morse potential energy model in D dimension, Eur. Phys. J. Plus, 129 (12), 264.

DOI: https://doi.org/10.22146/ijc.63294