Lie Algebraic Modeling of Dibromodichlorosilane Vibrational Frequencies Using Casimir and Majorana Operators
Resumo
Mathematical models used to analyse polyatomic molecule vibrational spectra must account for anharmonicity and molecular symmetry while being computationally efficient. We applied features of the Lie algebraic model relevant to the vibrational analysis of dibromodichlorosilane (SiBr₂Cl₂) and provided details. The vibrational Hamiltonian is explicitly formulated within the U(2) algebra with its Casimir and Majorana operators. The operator approach yields an algebraic representation of stretching and bending vibrations, eliminating the need for differential equations, thereby allowing for a direct matrix representation. Fundamental vibrational frequencies are computed and carried through to the second overtone. The results confirm that vibrational frequencies, within the algebraic structure, mathematically capture anharmonic and intermode effects. Additionally, the results show that the method is highly accurate compared to more conventional techniques and requires a minimal set of parameters. This study provides evidence of the value of Casimir and Majorana operators for constitutive algebraic models of vibrational frequencies and an important step towards broadening the scope of their use in molecular spectroscopy, mathematical physics, and operator theory.
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Referências
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25. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular vibrations: The theory of infrared and Raman vibrational spectra, Dover Publications, New York, 1980.
26. D. Papoušek and M. R. Aliev, Molecular vibrational–rotational spectra: Theory and applications of high-resolution infrared, microwave, and Raman spectroscopy, Elsevier, Amsterdam, 1982.
27. G. Herzberg, Infrared and Raman spectra of polyatomic molecules, Van Nostrand Reinhold, New York, 1945.
28. T. Shimanouchi, Tables of molecular vibrational frequencies consolidated, vol. 1, National Bureau of Standards, Washington, D.C., 1972, 1–60.
2. J. L. Dunham, The energy levels of a rotating vibrator, Physical Review 41 (1932), no. 6, 721–731. https://doi.org/10.1103/PhysRev.41.721.
3. M. Molski and J. Konarski, Acta Physica Polonica A 81 (1992), 495. https://doi.org/10.12693/APhysPolA.81.495.
4. M. Molski and J. Konarski, Modified Kratzer-Fues formula for rotation-vibration energy of diatomic molecules, Acta Physica Polonica A 82 (1992), no. 6, 927–936. https://doi.org/10.12693/APhysPolA.82.927.
5. P. Eshghi, L. Moafi, M. Alidoosti, and D. Nasr Esfahani, Colorimetric detection of fluoride and mercury (II) ions using isatin Schiff base skeleton bearing pyridine-2-carboxamidine moiety: Experimental and theoretical studies, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 305 (2024), 123467. https://doi.org/10.1016/j.saa.2023.123467.
6. C. Prabhu, P. Rajesh, M. Lawrence, S. Sahaya Jude Dhas, and A. I. Almansour, Computational aspects of DFT, HOMO-LUMO, PED, molecular docking and basic characterisations of Octadec-9-enoic acid (C18H34O2), Molecular Physics 123 (2024), no. 4. https://doi.org/10.1080/00268976.2024.2385572.
7. D. Bégué, P. Carbonnière, V. Barone, and C. Pouchan, Performance of ab initio and DFT PCM methods in calculating vibrational spectra in solution: Formaldehyde in acetonitrile as a test case, Chemical Physics Letters 416 (2005), no. 4–6, 206–211. https://doi.org/10.1016/j.cplett.2005.09.099.
8. A. Márquez, J. Fernández Sanz, M. Gelizé, and A. Dargelos, Ab initio calculations of molecular and electronic structure of disilane. I. Molecular force field and vibrational spectrum, Chemical Physics 149 (1991), no. 3, 311–318. https://doi.org/10.1016/0301-0104(91)90030-W.
9. R. Bala, V. S. Prasannaa, D. Chakravarti, D. Mukherjee, and B. P. Das, Ab initio spectroscopic studies of AlF and AlCl molecules, International Journal of Modern Physics B 39 (2024), no. 2, 2550024. https://doi.org/10.1142/S0217979225500249.
10. B. R. Westbrook and R. C. Fortenberry, Taming semi-empirical methods for PAHs and vibrational spectra, Journal of Molecular Spectroscopy 398 (2023), 111846. https://doi.org/10.1016/j.jms.2023.111846.
11. H. Chen, Q. Zhang, Z. Liu, and Y. Wang, Vibrational spectra and potential energy surfaces of small molecules, Chinese Physics B 33 (2024), no. 1, 013101. https://doi.org/10.1088/1674-1056/ad0d0b.
12. R. Lemus and A. Frank, Algebraic approach to vibrational spectra of tetrahedral molecules: Application to methane, Journal of Chemical Physics 101 (1994), no. 10, 8321–8332. https://doi.org/10.1063/1.468450.
13. F. Iachello and R. D. Levine, Algebraic theory of molecules, Oxford University Press, New York, 1995.
14. S. Teppala and V. Jaliparthi, Exploring cyclohexane vibrational dynamics through a Lie algebraic Hamiltonian framework, Ukrainian Journal of Physical Optics 25 (2024), no. 3, 03093–03100. https://doi.org/10.3116/16091833/Ukr.J.Phys.Opt.2024.03093.
15. S. Nallagonda and V. Jaliparthi, Higher overtone vibrational frequencies in naphthalene using the Lie algebraic technique, Ukrainian Journal of Physical Optics 25 (2024), no. 2, 02080–02085.
16. S. Oss, Algebraic models in molecular spectroscopy, Advances in Chemical Physics 93 (1966), 455. https://doi.org/10.1002/9780470141526.ch8.
17. J. Vijayasekhar, P. Suneetha, and K. Lavanya, Vibrational spectra of cyclobutane-d8 using symmetry-adapted one-dimensional Lie algebraic framework, Ukrainian Journal of Physical Optics 24 (2023), no. 3, 193–199.
18. V. Jaliparthi and A. S. Alali, Spectroscopic modelling of PCl₅ and SbCl₅ vibrations via Lie algebraic Hamiltonian with Casimir and Majorana operators, Ukrainian Journal of Physical Optics 26 (2025), no. 3, 03078–03087.
https://doi.org/10.3116/16091833/Ukr.J.Phys.Opt.2025.03078
19. R. Li, X. Gu, X. Lin, and Y. Wu, MRCI+Q calculations on electronic structure and spectroscopy of low-lying electronic states of silicon monobromide including spin–orbit coupling effect, Journal of Quantitative Spectroscopy and Radiative Transfer 329 (2024), 109207. https://doi.org/10.1016/j.jqsrt.2024.109207.
20. F. Mélen, I. Dubois, and H. Bredohl, New rotational analysis of the B′²Δ–X²Π transition of SiCl, Journal of Molecular Spectroscopy 139 (1990), no. 2, 361–368. https://doi.org/10.1016/0022-2852(90)90073-Y.
21. K. K. Irikura, Erratum: Experimental vibrational zero-point energies: Diatomic molecules [J. Phys. Chem. Ref. Data 36, 389–397 (2007)], Journal of Physical and Chemical Reference Data 38 (2009), no. 3, 749. https://doi.org/10.1063/1.3167794.
22. M. R. Balla and V. Jaliparthi, Vibrational Hamiltonian of methylene chloride using U(2) Lie algebra, Molecular Physics 119 (2020), no. 5.
https://doi.org/10.1080/00268976.2020.1828634.
23. F. A. Cotton, Chemical applications of group theory, 3rd ed., Wiley-Interscience, New York, 1990.
24. K. Nakamoto, Infrared and Raman spectra of inorganic and coordination compounds, Part A: Theory and applications in inorganic chemistry, 6th ed., Wiley, Hoboken, 2009.
25. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular vibrations: The theory of infrared and Raman vibrational spectra, Dover Publications, New York, 1980.
26. D. Papoušek and M. R. Aliev, Molecular vibrational–rotational spectra: Theory and applications of high-resolution infrared, microwave, and Raman spectroscopy, Elsevier, Amsterdam, 1982.
27. G. Herzberg, Infrared and Raman spectra of polyatomic molecules, Van Nostrand Reinhold, New York, 1945.
28. T. Shimanouchi, Tables of molecular vibrational frequencies consolidated, vol. 1, National Bureau of Standards, Washington, D.C., 1972, 1–60.
Publicado
2025-12-21
Seção
Mathematics and Computing - Innovations and Applications (ICMSC-2025)
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