Vol. 28, issue 05, article # 2

Nikitin A.V. Calculation of vibrational energy levels of symmetric molecules from potential energy surface. // Optika Atmosfery i Okeana. 2015. V. 28. No. 05. P. 379-390 [in Russian].
Copy the reference to clipboard

The algorithm of energy level calculation for symmetric 4–6 atomic molecules was considered. The question of choice of coordinates, of the form of potential energy surface, and methods of solutions of Schrödinger equation were considered. The kinetic energy operator for molecules of AB4 type in internal mass-independent coordinates was built.


vibrational energy levels, potential energy surface, operator of kinetic energy, infrared spectra, variational methods


  1. Partridge H., Schwenke D.W. The determination of an accurate isotope dependent potential energy surface for water from extensive ab initio calculations and experimental data // J. Chem. Phys. 1997. V. 106. P. 4618–4639.
  2. Schwenke D.W., Partridge H. Vibrational energy levels for CH4 from an ab initio potential // Spectrochim. Acta A. 2001. V. 57. P. 887.
  3. Schwenke D.W. Towards accurate ab initio predictions of the vibrational // Spectrochim. Acta A. 2002. V. 58. P. 849–861.
  4. Ovsyannikov R., Yurchenko S.N., Carvajal M., Thiel W., Jensen P. Vibrational energies of PH3 calculated variationally at the complete basis set limit // J. Chem. Phys. 2008. V. 129. P. 044309.
  5. Wang X.G., Carrington T., Jr. Contracted basis Lanczos methods for computing numerically exact rovibrational levels of methane // J. Chem. Phys. 2004. V. 121. P. 2937–2954.
  6. Wang X.G., Carrington T. A contracted basis-Lanczos calculation of vibrational levels of methane: Solving the Schrödinger equation in nine dimensions // J. Chem. Phys. 2003. V. 119. P. 101.
  7. Yurchenko S.N., Barber R.J., Tennyson J. A variationally computed line list for hot NH3 // Mon. Notic. Roy. Astron. Soc. 2011. V. 413. P. 1828–1834.
  8. Rothman L.S., Gordon I.E., Barbe A., Benner D.C., Ber-nath P.F., Birk M., Bizzocchi L., Boudon V., Brown L.R., Campargue A., Chance K., Cohen E.A., Coudert L.H., Devi V.M., Drouin B.J., Fayt A., Flaud J.-M., Gamache R.R., Harrison J.J., Hartmann J.-M., Hill C., Hodges J.T., Jacquemart D., Jolly A., Lamouroux J., Le Roy R.J., Li G., Long D.A., Lyulin O.M., Mackie C.J., Massie S.T., Mikhailenko S., Muller H.S.P., Naumenko O.V.,  Nikitin A.V., Orphal J., Perevalov V., Perrin A., Polovtseva E.R., Richard C., Smith M.A.H., Starikova E., Sung K., Tashkun S., Tennyson J., Toon G.C., Tyuterev Vl.G., Wagner G. The HITRAN 2008 molecular spectroscopic database // J. Quant. Spectrosc. Radiat. Transfer. 2009. V. 110. P. 533–600.
  9. Jacquinet-Husson N., Crepeau L., Armante R., Boutammine C., Chedin A., Scott N.A., Crevoisier C., Capelle V., Boone C., Poulet-Crovisier N., Barbe A., Campargue A., Benilan Y., Benner D.C., Bézard B., Boudon V., Brown L.R., Coudert L.H., Coustenis A., Dana V., Fally S., Fayt A., Flaud J.-M., Goldman A., Herman M., Harris G.J., Jacquemart D., Jolly A., Kleiner I., Kleinböhl A., Kwabia-Tchana F., Lavrentjeva N., Lacome N., Mandin J-Y., Maki A., Malathy Devi V., Mikhailenko S., Miller C.E., Moazzen-Ahmadi N., Nikitin A., Orphal J., Perevalov V., Perrin A., Petkie D.T., Predoi-Cross A., Rinsland C.P., Remedios J., Rotger M., Sung K., Tashkun S., Tennyson J., Toth R.A., Vandaele A.C., Vander Auwera J., Xu L.-H. The GEISA spectroscopic database: Current and future archive for Earth and planetary atmosphere studies // J. Quant. Spectrosc. Radiat. Transfer. 2008. V. 109. P. 1043–1059.
  10. Ogilvie J.F. Vibrational and Rotational Spectroscopy of Diatomic Molecules. Cambridge: Academic Press, 1998. 449 p.
  11. Watson J.K.G. The isotope dependence of diatomic Dunham coefficients // J. Mol. Spectrosc. 1980. V. 80, N 2. P. 411–421.
  12. Szalay P.G., Holka F., Fremont J., Rey M., Peterson K.A., Tyuterev V.G. Are ab initio quantum chemistry methods able to predict vibrational states up to the dissociation limit for multi-electron molecules close to spectroscopic accuracy // Phys. Chem. Chem. Phys. 2011. V. 13, N 9. P. 3654–3659.
  13. Huang X., Schwenke D.W., Tashkun S.A. An isotopic- independent highly accurate potential energy surface for CO2 isotopologues and an initial 12C16O2 infrared line list // J. Chem. Phys. 2012. V. 136. P. 124311.
  14. Lodi L., Tennyson J. Theoretical methods for small-molecule ro-vibrational spectroscopy // J. Phys. B. 2010. V. 43, N 13. P. 133001.
  15. Pavanello M., Adamowicz L., Alijah A., Zobov N., Mizus I.I., Polyansky O., Tennison J., Szidarovszky T., Csaszar A.G. Calibration-quality adiabatic potential energy surfaces for H3+ and its isotopologues // J. Chem. Phys. May 2012. V. 136, N 18. P. 184303.
  16. Polyansky O.L., Ovsyannikov R.I., Kyuberis A.A., Lo-di L., Tennyson J., Zobov N.F. Calculation of Rotation- Vibration Energy Levels of the Water Molecule with Near-Experimental Accuracy Based on an ab initio Potential Energy Surface // J. Phys. Chem. A. 2013. V. 117, N 39. P. 9633–9643.
  17. Yurchenko S.N., Carvajal M., Thiel W., Jensen P. Ab initio dipole moment and theoretical rovibrational intensities in the electronic ground state of PH3 // J. Mol. Spectrosc. 2006. V. 239, N 1. P. 71–87.
  18. Nikitin A.V., Holka F., Tyuterev V.G., Fremont J. Vibration energy levels of the PH3, PH2D, and PHD2 molecules calculated from high order potential energy surface // J. Chem. Phys. 2009. V. 131. P. 244312.
  19. Sousa-Silva C., Yurchenko S.N., Tennison J. A computed room temperature line list for phosphine // J. Mol. Spectrosc. 2013. V. 288. P. 28–36.
  20. Nikitin A.V., Rey M., Tyuterev V.G. High order dipole moment surfaces of PH3 and ab initio intensity predictions in the Octad range // J. Molec. Spectrosc. 2014. V. 305. P. 40–47.
  21. Huang X., Schwenke D.W., Lee T.J. Rovibrational spectra of ammonia. I. Unprecedented accuracy of a potential energy surface used with nonadiabatic corrections // J. Chem. Phys. 2011. V. 134, N 4. P. 044320.
  22. Huang X., Schwenke D.W., Lee T.J. Rovibrational spectra of ammonia. II. Detailed analysis, comparison, and prediction of spectroscopic assignments for 14NH3, 15NH3, and 14ND3 // J. Chem. Phys. 2011. V. 134, N 4. P. 044321.
  23. Marquardt R., Sagui K., Zheng J., Thiel W., Luckhaus D., Yurchenko S., Mariotti F., Quack M. Global Analytical Potential Energy Surface for the Electronic Ground State of NH3 from High Level ab initio Calculations // J. Phys. Chem. A. 2013. V. 117, N 32. P. 7502–7522.
  24. Polyansky O.L., Kozin I.N., Ovsyannikov R.I., Malyszek P., Koput J., Tennyson J., Yurchenko S.N. Variational calculation of highly excited rovibrational energy levels of H2O2 // J. Phys. Chem. A. 2013. V. 117, N 32. P. 7367–7377.
  25. Malyszek P., Koput J. Accurate ab initio potential energy surface and vibration-rotation energy levels of hydrogen peroxide // J. Comput. Chem. 2013. V. 34, N 5. P. 337–345.
  26. Yachmenev A., Yurchenko S.N., Jensen P., Walter T. A new “spectroscopic” potential energy surface for formaldehyde in its ground electronic state // J. Chem. Phys. 2011. V. 134. P. 244307.
  27. Urru A., Kozin I.N., Mulas G., Braams B.J., Tennyson J. Ro-vibrational spectra of C2H2 based on variational nuclear motion calculations // Mol. Phys. 2010. V. 108, N 15. P. 1973–1990.
  28. Martin J.M.L., Lee T.J., Taylor P.R. A purely ab initio spectroscopic quality quartic force field for acetylene // J. Chem. Phys. 1998. V. 108, N 2. P. 676–691.
  29. Wang X.G., Carrington T.J. Using experimental data and a contracted basis Lanczos method to determine an accurate methane potential energy surface from a least squares optimization // J. Chem. Phys. 2014. V. 141, N 15. P. 154106.
  30. Никитин А.В. Моделирование колебательных уровней энергии метана из ab initio поверхности потенциальной энергии // Оптика и спектроскопия. 2009. V. 106. P. 176.
  31. Rey M., Nikitin A.V., Tyuterev V.G. First principles intensity calculations of the methane rovibrational spectra in the infrared up to 9300 cm–1 // Phys. Chem. Chem. Phys. 2013. V. 15, N 25. P. 10049–10061.
  32. Yurchenko S.N., Tennyson J. ExoMol line lists-IV. The rotation-vibration spectrum of methane up to 1500 K // Mon. Notic. Roy. Astron. Soc. 2014. V. 440. P. 1649–1661.
  33. Yurchenko S.N., Tennyson J., Barber R.J., Thiel W. Vibrational transition moments of CH4 from first principles // J. Mol. Spectrosc. 2013. V. 291. P. 69–76.
  34. Rey M., Nikitin A.V., Tyuterev V.G. Theoretical hot methane line lists up to T = 2000 K for astrophysical applications // Astrophys. J. 2014. V. 788. P. 1.
  35. Manson S.A., Law M.M., Atkinson I.A., Thomson G.A. The molecular potential energy surface and vibrational energy levels of methyl fluoride. Part II // Phys. Chem. Chem. Phys. 2006. V. 8. P. 2855–2865.
  36. Nikitin A.V., Rey M., Tyuterev V.G. Rotational and vibrational energy levels of methyl fluoride calculated from a new potential energy surface // J. Mol. Spectrosc. 2012. V. 274. P. 28–34.
  37. Nikitin A.V. Vibrational energy levels of methyl chloride calculated from full dimensional ab initio potential energy surface // J. Mol. Spectrosc. 2008. V. 252. P. 17–22.
  38. Avila G., Carrington T.J. Using a pruned basis, a non-product quadrature grid, and the exact Watson normal-coordinate kinetic energy operator to solve the vibrational Schrödinger equation for C2H4 // J. Chem. Phys.  2011. V. 135, N 6. P. 064101.
  39. Carter S., Sharma A.R., Bowman J.M. First-principles calculations of rovibrational energies, dipole transition intensities and partition function for ethylene using MULTIMODE // J. Chem. Phys. 2012. V. 137, N 15. P. 154301.
  40. Delahaye T., Nikitin A., Rey M., Szalay P., Tyuterev V.G. A new accurate ground-state potential energy surface of ethylene and predictions for rotational and vibrational energy levels // J. Chem. Phys. 2014. V. 141. P. 104301.
  41. Carter S., Handy N.C., Bowman J.M. High torsional vibrational energies of H2O2 and CH3OH studied by MULTIMODE with a large amplitude motion coupled to two effective contraction schemes // Mol. Phys. 2009. V. 107. P. 727–737.
  42. Wang X.G., Carrington T. Vibrational energy levels of CH5+ // J. Chem. Phys. V. 129, N 23. P. 234102.
  43. Tennyson J., Yurchenko S. The Status of Spectroscopic Data for the Exoplanet Characterisation Missions // MNRAS. 2014. P. 1–12.
  44. Underwood D.S., Tennyson J., Yurchenko S.N. An ab initio variationally computed room-temperature line list for (SO3) // Phys. Chem. Chem. Phys. 2013. V. 15. P. 10118–10125.
  45. Bowman J.M., Carrington T., Meyer H.D. Variational quantum approaches for computing vibrational energies of polyatomic molecules // Mol. Phys. 2008. V. 106. P. 2145–2182.
  46. Makushkin Ju.S., Tjuterev V.G. Metody vozmushhenij i jeffektivnye gamil'toniany v molekuljarnoj spektroskopii. Novosibirsk: Nauka, 1984. 239 p.
  47. Kiselev A.A., Ljapcev A.V. Kvantovo-mehanicheskaja teorija vozmushhenij. Diagrammnyj metod. L.: Izdatelstvo LGU, 1989. 358 p.
  48. Tyuterev V.G., Tashkun S.A., Seghir H. High-Order Contact Transformations: General Algorithm, Computer Implementation and Triatomic Tests // Proc. SPIE. 2004. V. 5311. P. 164–175.
  49. Krasnoshchekov S.V., Isayeva E.V., Stepanov N.F. Numerical-Analytic Implementation of the Higher-Order Canonical Van Vleck Perturbation Theory for the Interpretation of Medium- Sized Molecule Vibrational Spectra // J. Phys. Chem. A. 2012. V. 116. P. 3691−3709.
  50. Cassam-Chenai P., Bouret Y., Rey M., Tashkun S.A., Nikitin A.V., Tyuterev V.G. Ab initio Effective Rotational Hamiltonians: A Comparative Study // Int. J. Quant. Chem. 2012. V. 136. P. 174309.
  51. Krasnoshchekov S.V., Craig N.C., Stepanov N.F. An-harmonic Vibrational Analysis of the Gas-Phase Infrared Spectrum of 1,1-Difluoroethylene Using the Operator Van Vleck Canonical Perturbation Theory // J. Phys. Chem. A. 2013. V. 117. P. 3041–3056.
  52. Tyuterev V.G., Tashkun S.A., Rey M., Kochanov R.V., Nikitin A.V., Delahaye T. Accurate spectroscopic models for methane polyads derived from a potential energy surface using high-order contact transformations // J. Phys. Chem. A. 2013. V. 117. P. 13779–13805.
  53. Nikitin A.V., Rey M., Tyuterev V.G. New dipole moment surfaces of methane // Chem. Phys. Lett. 2013. V. 565, N 5. P. 5–11.
  54. Manson S.A., Law M.M. General internal coordinate gradient vectors and the vibrational kinetic energy operator of centrally-connected penta-atomic systems. Part I // Phys. Chem. Chem. Phys. 2006. V. 8. P. 2848–2854.
  55. Nikitin A.V., Mikhailenko S., Morino I., Yokota T., Kumazawa R., Watanabe T. Isotopic substitution shifts in methane and vibrational band assignment in the 5560–6200 cm–1 region // J. Quant. Spectrosc. Radiat. Transfer. 2009. V. 110, N 12. P. 964–973.
  56. Rey M., Nikitin A.V., Tyuterev V.G. Predictions for methane spectra from potential energy and dipole moment surfaces: Isotopic shifts and comparative study of 13CH4 and 12CH4 // J. Mol. Spectrosc. 2013. V. 291. P. 85–97.
  57. Watson J.K.G. A comment on the use of redundant vibrational coordinates // J. Mol. Struct. 2004. V. 695–696. P. 71–75.
  58. Martinez-Torres E. Formulation of the vibrational theory in terms of redundant internal coordinates // J. Mol. Struct. 2000. V. 529. P. 53–61.
  59. Mills I.M. Redundant coordinates in harmonic force-field calculations // Chem. Phys. Lett. 1969. V. 3, N 5. P. 267–271.
  60. Huang S.W., Carrington T., jr. A new iterative method for calculating energy levels and wave functions // J. Chem. Phys. 2000. V. 112, N 20. P. 8765–8771.
  61. Wang X.G., Carringhton T., jr. Deficiencies of the bend symmetry coordinates used for methane // J. Chem. Phys. 2003. V. 118. P. 6260–6263.
  62. Papouśek D., Aliev M.R. Molecular Vibrational-rotational spectra. Amsterdam: Elsevier scientific publishing company, 1982. 314 p.
  63. Rey M., Nikitin A.V., Tyuterev V.G. Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: A method for computing energy levels from potential energy surfaces for symmetric-top molecules // Mol. Phys. 2010. V. 108. P. 2121–2135.
  64. Rey M., Nikitin A.V., Tyuterev V.G. Complete nuclear motion Hamiltonian in the irreducible normal mode tensor operator formalism for the methane molecule // J. Chem. Phys. 2012. V. 136, N 24. P. 244106.
  65. Law M.M., Duncan J.L., Mills I.M. The general harmonic force field of methyl fluoride // J. Mol. Structure. 1992. V. 260. P. 323–331.
  66. Duncan J.L., Mills I.M. The calculation of force constant and normal coordinates IV. XH4 and XH3 molecules // Spectrochim. Acta. 1994. V. 20, N 5. P. 523–546.
  67. Carter S., Culik S.J., Bowman J.M. Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu(100) // J. Chem. Phys. 1997. V. 107. P. 10458.
  68. Banker F. Simmetrija molekul i molekuljarnaja spektroskopija. M.: Mir, 1981. 444 p.
  69. Nikitin A.V. Algoritm vychislenija urovnej jenergii molekul tipa ABC3 i AB4 iz poverhnosti potencial'noj jenergii // Optika atmosf. i okeana. 2007. V. 20, N 9. P. 776–779.
  70. Zhilinskij B.I., Perevalov V.I., Tjuterev V.G. Metod neprivodimyh tenzornyh operatorov v teorii spektrov molekul. Novosibirsk: Nauka, 1987. 230 p.
  71. Marquardt R., Quack M. // J. Chem. Phys. 1998. V. 109. P. 10628.
  72. Demaison J. Experimental, semi-experimental and ab initio equilibrium structures // Mol. Phys. 2007. V. 105. P. 3109–3138.
  73. Nikitin A.V., Rey M., Tyuterev V.G. Rotational and vibrational energy levels of methane calculated from a new potential energy surface // Chem. Phys. Lett. 2011. V. 501. P. 179–186.
  74. Werner H.-J., Knowles P.J., Knizia G., Manby F.R., Schutz M., Celani P., Korona T., Lindh R., Mitrushenkov A., Rauhut G., Shamasundar K.R., Adler T.B., Amos R.D., Bernhardsson A., Berning A., Cooper D.L., Deegan M.J.O., Dobbyn A.J., Eckert F., Goll E., Hampel C., Hesselmann A., Hetzer G., Hrenar T., Jansen G., Koppl C., Liu Y., Lloyd A.W., Mata R.A., May A.J., McNicholas S.J., Meyer W., Mura M.E., Nicklaβ A., O'Neill D.P., Palmieri P., Peng D., Pfluger K., Pitzer R., Reiher M., Shiozaki T., Stoll H., Stone A.J., Tarroni R., Thorsteinsson T., Wang M. MOLPRO, version 2009.1, a package of ab initio programs. 2010.
  75. Cizek J. On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods // J. Chem. Phys. 1966. V. 45, N 11. P. 4256.
  76. Purvis G.D., Bartlett R.J. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples // J. Chem. Phys. 1982. V. 76. P. 1910.
  77. Dunning T.H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen // J. Chem. Phys. 1989. V. 90. P. 1007.
  78. Woon D.E., Dunning T.H. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon // J. Chem. Phys. 1993. V. 98. P. 1358.
  79. Knizia G., Adler T., Werner H. Simplified CCSD(T)-F12 methods: Theory and benchmarks // J. Chem. Phys. 2009. V. 130. P. 054104.
  80. Schwenke D.W., Partridge H. Vibrational energy levels for CH4 from an ab initio potential // Spectrochim. Acta A. 2001. V. 57. P. 887–895.
  81. Marquardt R., Quack M. Global Analytical Potential Hypersurface for Large Amplitude Nuclear Motion and Reactions in Methane II. Characteristic Properties of the Potential and Comparison to Other Potentials and Experimental Information // J. Phys. Chem. 2004. V. 108. P. 3166–3181.
  82. Watson J.K.G. Simplification of the molecular vibration-rotation hamiltonian // Mol. Phys. 1968. V. 15, N 5. P. 479.
  83. Tennyson J., Sutcliffe B.T. The ab initio calculation of the vibrational-rotational spectrum of triatomic systems in the close-coupling approach, with KCN and H2Ne as examples // J. Chem. Phys. 1982. V. 77, N 8. P. 4061–4072.
  84. Gatti F., Iung C., Menou M., Justum Y., Nauts A., Chapuisat X. Vector parametrization of the N-atom problem in quantum mechanics. I. Jacobi vectors // J. Chem. Phys. 1998. V. 108, N 21. P. 8804–8820.
  85. Gatti F., Iung C., Menou M., Chapuisat X. Vector parametrization of the N-atom problem in quantum mechanics. II. Coupled-angular-momentum spectral representations for four-atom systems // J. Chem. Phys.  1998. V. 108, N 21. P. 8821–8829.
  86. Iung C., Gatti F., Viel A., Chapuisat X. Vector parametrization of the N-atom problem in quantum mechanics with non-orthogonal coordinates // Phys. Chem. Chem. Phys. 1999. V. 1. P. 3377–3385.
  87. Schwenke D.W. New rovibrational kinetic energy operators using polyspherical coordinates for polyatomic molecules // J. Chem. Phys. 2003. V. 118, N 23. P. 10431–10438.
  88. Mladenović M. Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parametrization. I. Orthogonal representations // J. Chem. Phys. 2000. V. 112. P. 1070–1081.
  89. Mladenović M. Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parametriza-tion. II. Nonorthogonal descriptions of internal molecular geometry // J. Chem. Phys. 2000. V. 112, N 3. P. 1082–1095.
  90. Lukka T. A simple method for the derivation of exact quantum-mechanical vibration-rotation Hamiltonians in terms of internal coordinates // J. Chem. Phys. 1994. V. 102, N 10. P. 3945–3955.
  91. Xie J., Tennyson J. Variational calculations of vibra-tional energy levels for XY4 molecules: 1. Stretching states // Mol. Phys. 2002. V. 100, N 10. P. 1615–1622.
  92. Xie J., Tennyson J. Variational calculations of vibrational energy levels for XY4 molecules: 2. Bending states of methane // Mol. Phys. 2002. V. 100, N 10. P. 1623–1632.
  93. Nikitin A.V. Operator kolebatel'noj kineticheskoj jenergii dlja molekul tipa AB4 // Optika atmosf. i okeana. 2002. V. 15, N 9. P. 722–726.
  94. Nikitin A.V., Rey M., Tyuterev V.G. An efficient method for energy levels calculation using full symmetry and exact kinetic energy operator: Tetrahedral molecules // J. Chem. Phys. 2015. V. 142, N 9. P. 094118.
  95. Csaszar A.G., Handy N.C. Exact quantum mechanical vibrational kinetic energy operator of sequentially bonded molecules in valence internal coordinates // J. Chem. Phys. 1995. V. 102, N 5. P. 3962–3967.
  96. Poirier B. Exploiting both C3v symmetry and sparsity in vibrational calculations for methanelike molecules // J. Chem. Phys. 2003. V. 119. P. 90.
  97. Nikitin A.V. New efficient algorithm for the calculation of energy levels of AB3 type molecules // Mol. Phys. 2011. V. 109. P. 483–492.
  98. Nikitin A.V. Modelirovanie kolebatel'nyh urovnej jenergii metana iz ab initio poverhnosti potencial'noj jenergii // Opt. Spectros. 2009. V. 106, N 2. P. 176.
  99. Light J.C., Hamilton I.P., Lill J. Generalized discrete variable approximation in quantum mechanics // J. Chem. Phys. 1984. V. 82, N 3. P. 1400–1409.
  100. Light J.C., Carrington T., jr. Discrete-variable representations and their utilization // Adv. Chem. Phys. 2000. V. 114. P. 263–310.
  101. Wang X.G., Carrington T., jr. A discrete variable representation method for studying the rovibrational quantum dynamics of molecules with more than three atoms // J. Chem. Phys. 2003. V. 118, N 15. P. 6946–6956.
  102. Matyus E., Simunek J., Csaszar A.G. On the variational computation of a large number of vibrational energy levels and wave functions for medium-sized molecules // J. Chem. Phys. 2009. V. 131, N 7. P. 074106.
  103. Csaszar A.G., Fabri C., Szidarovszky T., Matyus E., Furtenbacher T., Czako G. The fourth age of quantum chemistry: Molecules in motion // Phys. Chem. Chem. Phys. 2012. V. 14, N 3. P. 1085–1106.
  104. Wang X.G., Carrington T., jr. A finite basis representation Lanczos calculation of the bend energy levels of methane // J. Chem. Phys. 2003. V. 118, N 15. P. 6946–6956.
  105. Wang X.G., Carrington T., jr. Computing rovibrational levels of methane with curvilinear internal vibrational coordinates and an Eckart frame // J. Chem. Phys. 2013. V. 138. P. 104106.
  106. Wang X.G., Sibert III E.L. A nine-dimensional perturbative treatment of the vibrations of methane and its isotopomers // J. Chem. Phys. 1999. V. 111. P. 4510–4522.
  107. Yu H.G. An exact variational method to calculate vibrational energies of five atom molecules beyond the normal mode approach // J. Chem. Phys. 2002. V. 117. P. 2030–2037.
  108. Yu H.G. Converged quantum dynamics calculations of vibrational energies of CH4 and CH3D using an ab initio potential // J. Chem. Phys. 2004. V. 121. P. 6334–6340.
  109. Matyus E., Simunek J., Csaszar A. On the variational computation of a large number of vibrational energy levels and wave functions for medium-sized molecules // J. Chem. Phys. 2009. V. 131. P. 074106.
  110. Mandelshtam V.A., Taylor H.S. A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time signals // J. Chem. Phys. 1997. V. 106, N 12. P. 5085–5090.
  111. Cullum J.K., Willoughby R.A. Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Boston: Birkhauser, 1985. 294 p.
  112. Carrington T. Encyclopedia of Computational Chemistry. V. 5. New York: Wiley, 1998. P. 3157–3166.
  113. Yurchenko S.N., Tennyson J., Bailey J., Hollis M.D.J., Tinetti G. Spectrum of hot methane in astronomical objects using a comprehensive computed line list // Proc. National Academy of Sciences of the United States of America. 2014. V. 111, N 26. P. 9379–9383.
  114. Rey M., Nikitin A.V., Tyuterev V.G. Accurate first-principles calculations for 12CH3D infrared spectra from isotopic and symmetry transformations // J. Chem. Phys. 2014. V. 141, N 4. P. 044316.
  115. Carter S., Bowman J.M., Handy N.C. Extensions and tests of “multimode”: A code to obtain accurate vibra-tion/rotation energies of many-mode molecules // Theor. Chem. Accounts. 1998. V. 100, N 1–4. P. 191–198.
  116. Carter S. Variational calculations of rovibrational energies of CH4 and isotopomers in full dimensionality using an ab initio potential // J. Chem. Phys. 1999. V. 110. P. 8417–8423.
  117. Carter S., Bowman J.M. Variational calculations of rotational-vibrational energies of CH4 and isotopomers using an adjusted ab initio potential // J. Phys. Chem. A. 2000. V. 104, N 11. P.2355–2361.
  118. Wu J., Huang X., Carter S., Bowman J.M. Tests of MULTIMODE calculations of rovibrational energies of CH4 // J. Chem. Phys. 2006. V. 426. P. 285–289.
  119. Yurchenko S.N., Thiel W., Jensen P. Theoretical ROVib-rational Energies (TROVE): A robust numerical approach to the calculation of rovibrational energies for polyatomic molecules // J. Mol. Spectrosc. 2007. V. 245. P. 126–140.
  120. Yurchenko S.N., Carvajal M., Thiel W., Jensen P. // J. Mol. Spectrosc. 2006. V. 239. P. 71.
  121. Eckart C. Some Studies Concerning Rotating Axes and Polyatomic Molecules // Phys. Rev. 1935. V. 47. P. 552–558.
  122. Lehoucq R.B., Gray S.K., Zhang D.H., Light J.C. Vibrational eigenstates of four-atom molecules: A parallel strategy employing the implicitly restarted Lanczos method // Comput. Phys. Commun. 1998. V. 109, N 1. P. 15–26.
  123. Kozin I.N., Law M.M., Tennyson J., Hutson J.M. Calculating energy levels of isomerizing tetra-atomic molecules. II. The vibrational states of acetylene and vinylidene // J. Chem. Phys. V. 122, N 6. P. 064309.
  124. Pavlyuchko A.I., Yurchenko S.N., Tennyson J. Hybrid variational-perturbation method for calculating ro-vibrational energy levels of polyatomic molecules // J.  Chem. Phys. 2015. V. 142, N 9. P. 094309.
  125. Qu C., Prosmiti R., Bowman J.M. MULTIMODE calculations of the infrared spectra of H7 + and D7 + using ab initio potential energy and dipole moment surfaces // Theor. Chem. Accounts. 2013. V. 132, N 102. P. 1–7.