Vol. 34, issue 04, article # 1

Bykov A. D., Voronin B. A., Dudaryonok A. S., Polovtseva E. R. Shift of vibrational bands upon isotope substitution in molecules. // Optika Atmosfery i Okeana. 2021. V. 34. No. 04. P. 237–244. DOI: 10.15372/AOO20210401 [in Russian].
Copy the reference to clipboard

Abstract:

High-order Rayleigh–Schrödinger perturbation theory and series summation by the Padé–Hermit approximants are used to calculate the vibrational energy levels of isotopically substituted molecules. As an example, the levels of water isotopologues are calculated. It is found that the perturbation series diverge in the cases of both hydrogen by deuterium or tritium substitution and "heavy" oxygen atom (16О–18О) substitution. However, the use of the Padé–Hermit approximants makes it possible to calculate the isotopic level shifts quite accurately for any isotope substitution, both preserving the symmetry of the molecule and reducing it.

Keywords:

isotopic shift of vibrational levels, Rayleigh–Schrödinger perturbation theory, summation of divergent series of perturbation theory

References:

1. Gertsberg G. Kolebatel'nye i vrashchatel'nye spektry mnogoatomnyh molekul. M.: IL, 1949. 648 p.
2. Sverdlov L.M., Kovner M.A., Krajnov E.P. Kolebatel'nye spektry mnogoatomnyh molekul. M.: Nauka, 1970. 559 p.
3. Bykov A.D., Makushkin Yu.S., Ulenikov O.N. Izotopozameshchenie v mnogoatomnyh molekulah. Novosibirsk: Nauka, 1985. 157 p.
4. Bykov A.D., Makushkin Yu.S., Ulenikov O.N. On the displacements of the centers of vibration-rotation bands under isotope ubstitution in polyatomic molecules // J. Mol. Spectrosc. 1982. V. 93, N 1. P. 46–54.
5. Bykov A.D., Makushkin Yu.S., Ulenikov O.N. On the displacements of the centers of vibration-rotation lines under isotope substitution in polyatomic molecules // J. Mol. Phys. 1984. V. 51, N 4. P. 907–918.
6. Fowler P.W. Perturbation calculation of isotope shifts in molecular properties // Mol. Phys. 1983. V. 48, N 1. P. 153–160.
7. Arteca G.A, Fernández F.M., Castro E.A. Theory and Summation Methods in Quantum Mechanics // Large order perturbation theory and summation methods in quantum mechanics Large Order Perturbation (Lecture Notes in Chemistry). Berlin: Springer, 1990. 580 p.
8. Suslov I.M. Raskhodyashchiesya ryady teorii vozmushchenij // ZhETF. 2005. V. 127, iss. 6. P. 1350–1402.
9. Makushkin Yu.S., Terent'ev A.V., Ulenikov O.N. v kn. Molekulyarnaya spektroskopiya vysokogo i sverhvysokogo razresheniya. Novosibirsk: Nauka. 1976. P. 3–52.
10. Aliev M.R., Papousek D. Molecular Vibrational-Rotational Spectra. Elsevier, New York, 1982. 323 p.
11. Goodson D.Z. Resummation methods // Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012. V. 2, iss. 5. P. 743–761. DOI: 10.1002/wcms.92.
12. Sergeev A.V. Summation of the eigenvalue perturbation series by multi-valued Pade approximants: Application to resonance problems and double wells // J. Phys. A: Math. Gen. 1995. V. 28. P. 4157–4162.
13. Goodson D.Z., Sergeev A.V. On the use of algebraic approximants to sum divergent series for Fermi resonances in vibrational spectroscopy // J. Chem. Phys. 1999. V. 110 P. 8205–8206. DOI: 10.1063/1.478722.
14. Goodson D.Z. On the use of quadratic approximants to model diatomic potential energy curves // J. Mol. Phys. 2012. V. 110. N 15–16. P. 1681–1691.
15. Fernández F.M., Diaz C.G. Accurate summation of the perturbation series for periodic eigenvalue problems // Eur. Phys. J. D. 2001. V. 15 P. 41–46.
16. Duchko A.N., Bykov A.D. Resummation of divergent perturbation series: application to the vibrational states of H2CO molecule // J. Chem. Phys. 2015. V. 143. P. 154102.
17.  Bykov A.D., Duchko A.N. Multivalued property of Rayleigh–Schrödinger perturbation series for vibrational energy levels of molecules // Phys. Scr. 2019. V. 94, N 10. P. 105402. DOI: 10.1088/1402-4896/ab29fe.
18. Bykov A.D., Kalinin K.V., Duchko A.N. Calculation of vibrational HDO energy levels: Analysis of perturbation theory series // Opt. Spectrosc. 2013. V. 114. P. 359–367.
19. Krasnoshchekov S.V., Dobrolyubov E.O., Chan S. Fundamental'nyj analiz singulyarnyh i rezonansnyh yavlenij v kolebatel'nyh poliadah molekuly diftorsililena // Opt. i spektrosk. 2020. V. 128, N 12. P. 1795–1805.
20. Krasnoshchekov S.V., Dobrolyubov E.O., Syzgantseva M.A., Palvelev R.V. Rigorous vibrational Fermi resonance criterion revealed: Two different approaches yield the same result // J. Mol. Phys. 2020. V. 118, iss. 11. P. e1743887. DOI: 10.1080/00268976.2020. 1743887.
21. Krasnoshchekov S.V., Dobrolyubov E.O., Chang X. Hypoorous acid (HOF): A molecule with a rare 1-21 vibrational resonance and (8,3,2) polyad structure revea­led by Pade–Hermite resummation of divergent Raleigh – Schrödinger perturbation theory series // J. Quant. Spectrosc. Radiat. Transfer. 2021. V. 268. P. 107620-1–107620-15. DOI: 10.1016/j.jqsrt.2021.107620.
22. Mills I.M. Special Periodical Reports, Theoretical Chemistry. The Chemical Society, London, 1974. V. 1. P. 110.
23.  Tennyson J., Bernath P.F., Brown L.R., Campargue A., Császár A.G., Daumont L., Gamache R.R., Hodges J.T., Naumenko O.V., Polyansky O.L., Roth­man L.S., Vandaele A.C., Zobov N.F., Dénes N., Fazliev A.Z., Furtenbacher T., Gordon I.E., Hum S.-M., Szidarovszky T., Vasilenko I.A. IUPAC critical evaluation of the rotational–vibrational spectra of water vapor. Part IV. Energy levels and transition wavenumbers for D216O, D217O, and D218O // J. Quant. Spectrosc. Radiat. Transfer. 2014. V. 142. P. 93–108.
24. Tennyson J., Bernath P.F., Brown L.R., Campargue A., Császár A.G., Daumont L., Gamache R.R., Hodges J.T., Naumenko O.V., Polyansky O.L., Roth­man L.S., Toth R.A., Vandaele A.C., Zobov N.F., Fallym S., Fazliev A.Z., Furtenbacher T., Gordon I.E., Hun S.-M., Mikhailenko S.N., Voronin B.A. IUPAC critical evaluation of the rotational–vibrational spectra of water vapor. Part II. Energy levels and transition wave numbers for HD16O, HD17O, and HD18O // J. Quant. Spectrosc. Radiat. Transfer. 2010. V. 111. P. 2160–2184.
25. Tennyson J., Bernath P.F., Brown L.R., Campargue A., Carleer M.R., Csa´sza´r A.G., Gamache R.R., Hodges J.T., Jenouvrier A., Naumenko O.V., Polyan­sky O.L., Rothman L.S., Toth R.A., Vandaele A.C., Zobov N.F., Daumont L., Fazliev A.Z., Furtenbacher T., Gordon I.E., Mikhailenko S.N., Shirin S.V. IUPAC critical evaluation of the rotational–vibrational spectra of water vapor. Part I. Energy levels and transition wavenumbers for H217O and H218O // J. Quant. Spectrosc. Radiat. Transfer. 2009. V. 110. P. 573–596.
26. Mikhailenko S.N., Babikov Yu.L., Golovko V.F. Informatsionno-vychislitel'naya sistema «Spektroskopiya atmosfernyh gazov». Struktura i osnovnye funktsii // Optika atmosf. i okeana. 2005. V. 18, N 9. P. 765–776.
27. Down M.J., Tennyson J., Hara M., Hatano Y., Kobayashi K. Analysis of a tritium enhanced water spectrum between 7200 and 7245 cm-1 using new variational calculations // J. Mol. Spectrosc. 2013. V. 289. P. 35–40.
28. Schwenke D.W. First principles prediction of isotopic shifts in H2O // J. Chem. Phys. 2003. V. 118. P. 6898–6904. DOI: 10.1063/1.1561053.