Do we Really Know How to Define Normal Vibrations in Non-Rigid Molecular Systems?

Part of the Topics in Molecular Organization and Engineering book series (MOOE, volume 12)


The paper analyses ambiguities which exist both in a choice of slowly deforming geometries serving as reference points for vibrations in floppy molecules and in a definition of the large-amplitude internal variables near the selected reference configurations. Since the cited ambiguities change both the frequencies and the shape of normal vibrations, the vibrational analysis becomes an ill-defined problem even in the harmonic approximation. The author formulates some general criteria for selecting a“preferable”coordinate system, which would make it possible to give an intrinsic meaning to the term“harmonic vibrations of a floppy molecule”.


Internal Rotation Internal Variable Normal Vibration Harmonic Vibration Reference Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    A. Say vetz: “The kinetic energy of polyatomic molecules”, J. Chem. Phys. 7(1939) 383–389.ADSCrossRefGoogle Scholar
  2. [2]
    C. Eckart: “Some studies concerning rotating axes and polyatomic molecules”, Phys. Rev. 47 (1935) 552–558.ADSCrossRefGoogle Scholar
  3. [3]
    J.T. Hougen, P.R. Bunker and J.W.C. Johns: “The vibration-rotation problem in triatomic molecules allowing for a large-amplitude bending vibration”, J. Molec. Spectrosc. 34 (1970) 136–172.ADSCrossRefGoogle Scholar
  4. [4]
    G.A. Natanson: Modelling of nuclear motions in calculations of spectra of“non-rigid” molecules, Ph.D. Thesis, Leningrad, U.S.S.R. (1974 in Russian).Google Scholar
  5. [5]
    M.N. Adamov and G.A. Natanson: Separation of variables describing motions of nuclei and electrons in“non-rigid”molecules, Reprint of Inst. Theor. Phys. Acad. Sci. Ukr. SSR, ITF-76-82R, Kiev, USSR, (1976 in Russian).Google Scholar
  6. [6]
    G.L. Hofacker: “Quantentheorie chemischer Reactionen”, Z. Naturforsch A18 (1963) 607-619.ADSGoogle Scholar
  7. [7]
    R.A. Marcus: “Local approximation of potential energy surfaces permitting separation of variables”, J. Chem. Phys. 41 (1964) 610-616.MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    K. Fukui: “The charge and spin transfer in chemical reaction paths”, in The World of Quantum Chemistry, R. Dsudel, B. Pullman and D. Holland (Eds.), Reidel Publish. Co. (1974) p. 113.Google Scholar
  9. [9]
    G.A. Natanson,“Internal motion of a non-rigid molecule and its relation to the reaction path”, Mol Phys. 46 (1982) 481-512.Google Scholar
  10. [10]
    D.J. Rowe and A. Ryman: “Valleys and fall lines on a Riemannian manifold”, J. Math. Phys. 23 (1982) 732–735.MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. [11]
    S. Kato, H. Kato and K. Fukui: “A theoretical treatment on the behavior of the hydrogen-bonded proton in malonaldehyde”, J. Amer. Chem. Soc. 99 (1977) 684-691.CrossRefGoogle Scholar
  12. [12]
    M.V. Basilevsky: “Modern development of the reaction coordinate concept”, J. Molec. Struct. (Theochem) 103 (1983) 138–152.CrossRefGoogle Scholar
  13. [13]
    H.M. Pickett: “Vibration-rotation interactions and the choice of rotating axes for polyatomic molecules”, J. Chem. Phys. 56 (1972) 1715-1723.ADSCrossRefGoogle Scholar
  14. [14]
    G.A. Natanson: “Optimum choice of the reaction coordinate for adiabatic calculations of the tunnelling probabilities”, Chem. Phys. Lett. 190 (1992) 209–214.ADSCrossRefGoogle Scholar
  15. [15]
    N.R. Walet, A. Klein and G. Do Dang: “Reaction path and generalized valley approximation”, J. Chem. Phys. 91 (1989) 2848-2858.ADSCrossRefGoogle Scholar
  16. [16]
    N. Shida, J.E. Almöf and P.F. Barbara: “Molecular vibrations in a gradient extremal path”, Theor. Chim. Acta 76 (1989) 7–31.CrossRefGoogle Scholar
  17. [17]
    D.K. Hoffman, R.S. Nord and K. Ruedenberg: “Gradient extremals”, Theor. Chim. Acta 69 (1986) 265–279.CrossRefGoogle Scholar
  18. [18]
    M.V. Basilevsky and A.G. Shamov: “The local definition of the optimum ascent path on a multi-dimensional potential energy surface and its practical application for the location of saddle points”, Chem. Phys. 60 (1981) 347–358.CrossRefGoogle Scholar
  19. [19]
    G.A. Natanson: “A reduction of the reaction path formalism to the space of internal variables”, Chem. Phys. Lett. 178 (1991) 49–54.ADSCrossRefGoogle Scholar
  20. [20]
    C.R. Quade: “The interaction of a large amplitude internal motion with other vibrations in molecules. The effective Hamiltonian for the large amplitude motion”, J. Chem. Phys. 65 (1976) 700-705.ADSCrossRefGoogle Scholar
  21. [21]
    J.R. Villarreal, L.E. Bauman, and J. Laane: “Ring-puckering vibrational spectra of cyclopentene-d1 andcyclopentene-l,2,3,3-d4”,J. Phys. Chem. 80(1976) 1172–1177.CrossRefGoogle Scholar
  22. [22]
    L.E. Bauman, P.M. Killough, J.M. Cooke, J.R. Villarreal and J. Laane: “Two-dimensional potential energy surface for the ring puckering and ring twisting of cyclopentene-do,-1-d1,-l,2,3,3-d4, and-d8”, J. Phys. Chem. 86 (1982) 2000-2006.CrossRefGoogle Scholar
  23. [23]
    G.A. Natanson: “A new definition of the reaction coordinate via adiabatic dividing surfaces formed by classical trajectories”, Chem. Phys. Lett. 190 (1992) 215-224.ADSCrossRefGoogle Scholar
  24. [24]
    G.A. Natanson: “On invariance of localized Hamiltonians under feasible elements of the nuclear permutation-inversion group”, Adv. Chem. Phys. 58 (1985) 55-126.CrossRefGoogle Scholar
  25. [25]
    R.T. Skodje, D.G. Truhlar and B.C. Garrett: “A general small-curvature approximation for transition-state-theory transmission coefficients”, J. Phys. Chem. 85 (1981) 3019-3023.CrossRefGoogle Scholar
  26. [26]
    R.A. Marcus: “On the analytical mechanics of chemical reactions. Quantum mechanics of linear collisions”, J. Chem. Phys. 45 (1966) 4493-499.ADSCrossRefGoogle Scholar
  27. [27]
    R.A. Marcus: “On the analytical mechanics of chemical reactions. Classical mechanics of linear collisions”, J. Chem. Phys. 45 (1966) 4500–4504.ADSCrossRefGoogle Scholar
  28. [28]
    G. Do Dang, A. Bulgac and A. Klein: “Determination of the collective Hamiltonian in a self-consistent theory of large-amplitude adiabatic motion”, Phys. Rev. 36 (1987) 2661–2671.ADSGoogle Scholar
  29. [29]
    R.R. Newton and L.H. Thomas: “Internal molecular motions of large amplitude illustrated by the symmetrical vibration of ammonia”, J. Chem. Phys. 16 (1948) 310-319.ADSCrossRefGoogle Scholar
  30. [30]
    G. A. Natanson: “A choice of the semirigid bender model of the water molecule”, J. Molec. Spectrosc. 95 (1982) 63–67.ADSCrossRefGoogle Scholar
  31. [31]
    F.B. Brown, S.C. Tucker and D.G. Truhlar: “Semiclassical reaction-path methods applied to calculate the tunnelling splitting in ammonia”, J. Chem. Phys. 83 (1985) 4451-4455.ADSCrossRefGoogle Scholar
  32. [32]
    P. Bopp, D.R. McLaughlin and M. Wolfsberg: “Variational calculations of the lower vibra-tional energy levels of the ammonia molecule”, Z. Naturforsch. 37a (1982) 398–400.ADSGoogle Scholar
  33. [33]
    B. Maessen, P. Bopp, D.R. McLaughlin and M. Wolfsberg: “An improved variational calculations of the lower vibrational energy levels of the ammonia molecule”, Z. Naturforsch. 39a (1984) 1005–1006.ADSGoogle Scholar
  34. [34]
    A. Klein, N.R. Walet and G. Do Dang: “Classical theory of collective motion in the large amplitude, small velocity regime”, Ann. Rev. 208 (1991) 90-148.MathSciNetGoogle Scholar
  35. [35]
    G.A. Natanson and M.N. Adamov: “Hamiltonian of a polyatomic molecule. II. Eckart-Sayvetz conditions”, Vestn. Leningr. Univ. 10, (1974, in Russian) 24–32.Google Scholar
  36. [36]
    G.A. Natanson: “Comment on: Decoupling of the local mode stretching vibrations of water through rotational excitation. I. Quantum mechanics”, J. Chem. Phys. 88 (1988) 7252-7253.ADSCrossRefGoogle Scholar
  37. [37]
    E.B. Wilson Jr., J.C. Decius and P.C. Cross: Molecular Vibrations, McGraw-Hill, New York, (1955).Google Scholar
  38. [38]
    A.R. Hoy and P.R. Bunker: “The effective rotation-bending Hamiltonian of a triatomic molecule, and its application to extreme centrifugal distortion in the water molecule”, J. Molec. Spectrosc. 52 (1974) 439-456.ADSCrossRefGoogle Scholar
  39. [39]
    P. Jensen: “A new morse oscillator-rigid bender internal dynamics (MORBID) Hamiltonian for triatomic molecules”, J. Molec. Spectrosc. 128 (1988) 478-501.ADSCrossRefGoogle Scholar
  40. [40]
    D.H. Cress and C.R. Quade: “The interaction of inversion with other vibrations in ammonia”, J. Chem. Phys. 67 (1977) 5695–5701.ADSCrossRefGoogle Scholar
  41. [41]
    C.R. Quade: “Contributions of the interaction of internal rotation with other vibrations in the effective potential energy for internal rotation in molecules with symmetric internal rotors”, J. Chem. Phys. 73 (1980) 2107–2114.ADSCrossRefGoogle Scholar
  42. [42]
    Y. Guan and C.R. Quade: “Curvilinear coordinate formulation for vibration-rotation large-amplitude internal motion interactions. I. The general theory”, J. Chem. Phys. 84 (1986) 5624-5638.ADSCrossRefGoogle Scholar
  43. [43]
    Y. Guan and C.R. Quade: “Curvilinear coordinate formulation for vibration-rotation large-amplitude internal motion interactions. II. Application to the water molecule”, J. Chem. Phys. 86 (1987) 4808-4823.ADSCrossRefGoogle Scholar
  44. [44]
    H.C. Longuet-Higgins: “The symmetry groups of non-rigid molecules”, Mol. Phys. 15 (1963) 445-60.MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    P.R. Bunker: Molecular Symmetry and Spectroscopy, Academic Press, New York (1979).Google Scholar
  46. [46]
    J.T. Hougen: “Vibrational motions in dimethylacetylene”, Can. J. Phys. 43 (1965) 935-954.ADSCrossRefGoogle Scholar
  47. [47]
    P.R. Bunker: “Dimethylacetylene: An analysis of the theory required to interpret its vibrational spectrum”, J. Chem. Phys. (1967) 47 718–739.ADSCrossRefGoogle Scholar
  48. [48]
    M.N. Adamov and G.A. Natanson: “New aspects in the symmetry theory of polyatomic molecules”, Fiz. Molek. 2 (1976) 3–16 [see Los Alamos Translation LA-TR-77-32].Google Scholar
  49. [49]
    G.A. Natanson: “Classification of normal vibrations in nonrigid molecules”, Fiz. Molek. 6 (1978) 3–32.Google Scholar
  50. [50]
    G.A. Natanson: “On symmetry classification of normal vibrations in molecules with internal rotation”, in Symmetries and Properties of Non-Rigid Molecules: A Comprehensive Survey, by J. Maruani and J. Serre (Eds.), Studies in Phys. & Theor. Chem., Vol. 23 (1983), pp. 201–218.Google Scholar
  51. [51]
    A.J. Merer and J.K.G. Watson: “Symmetry considerations for internal rotation in ethylene-like molecules”, J. Molec. Spectrosc. 47 (1973) 499-514.ADSCrossRefGoogle Scholar
  52. [52]
    P. Groner: Gas-und Matrixinfrarotspektren von Nitroethan-Isotopen und der Einfluss der Internen Rotation derNitrogruppe, Ph.D. Thesis, Diss. No. 5394, Swiss Fed. Inst. Technology, Zürich, Switzerland (1974).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Chemical Dynamics CorporationUpper MarlboroUSA

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