Dynamics of Non-Rigid Molecules: The Exploration of Phase Space Via Resonant and Sub-Resonant Coupling

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


In systems of non-linear coupled oscillators, it is possible to specify initial distributions of excitation energy that cause the zero order frequencies of the oscillators to be in resonance. In the presence of coupling, these resonances mediate the exchange of energy between the zero order modes. The magnitude of this energy exchange is a measure of the width of the resonance. We show that for higher order resonances there exist sub-resonances that in other parts of phase space correspond to lower order resonances. These sub-resonances provide a mechanism for large amplitude, high frequency oscillatory excursions in phase space. They also induce sub-resonant modulation of the resonance widths. To analyse the dynamics of coupled non-linear oscillators, conventional methods typically treat the coupling as a small perturbation and“average”over rapidly oscillating non-resonant terms. This averaging neglects the effect of the sub-resonances which, for physically realistic couplings can have amplitudes as large or larger than the resonance. We discuss an alternative approach for the perturba-tive analysis of coupled non-linear oscillators designed to incorporate the effect of sub-resonances on the dynamics. We then show that for small values of coupling when first order perturbation theory is valid, the sub-resonances lead to new, nearly separable modes. However, for large coupling, sub-resonant effects cannot be transformed away and dominate phase space dynamics.


Action Space Resonance Width Resonance Centre Fourier Series Expansion Resonance Surface 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Grant Cooper (Ed.): From Cardinals to Chaos, Cambridge University Press, Cambridge (1989).Google Scholar
  2. [2]
    B.V. Chirikov: Phys. Rep. 52 (1979) 263.MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    G.M. Zaslavsky: Phys. Rep. 80 (1981) 157; R.Z. Sagdeev, D.A. Usikov and G.M. Zaslavski: Nonlinear Physics From the Pendulum to Turbulence and Chaos, Harwood Academic Publishers, New York (1988).MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    A.J. Lichtenberg and M.A. Lieberman: Regular and Stochastic Motion, Springer-Verlag, New York (1983).zbMATHGoogle Scholar
  5. [5]
    M. Tabor: Chaos and Integrability in Nonlinear Dynamics, Wiley, New York (1989); S.N. Rasband: Chaotic Dynamics of Nonlinear Systems, Wiley, New York (1989).zbMATHGoogle Scholar
  6. [6]
    J. Ford: Adv. Chem. Phys. 24 (1973) 155.CrossRefGoogle Scholar
  7. [7]
    M.V. Berry and M. Tabor: Proc. R. Soc. Lond. A349 (1976) 101.MathSciNetADSGoogle Scholar
  8. [8]
    R.A. Marcus: Discussions Faraday Soc. 55 (1973) 34; D.M. Wardlaw and R.A. Marcus: Adv. Chem. Phys. 70 (1987) 231; D.W. Noid, M.L. Koszykowski and R.A. Marcus: Annu. Rev. Phys. 32(1981)267.CrossRefGoogle Scholar
  9. [9]
    S.A. Rice: Adv. Chem. Phys. 47 (1981) 117; P. Brumer: Adv. Chem. Phys. 47 (1981) 201.CrossRefGoogle Scholar
  10. [10]
    J. Jortner, R.D. Levine and B. Pullman (Eds.): Mode Selective Chemistry, Kluwer, Dordrecht (1991); E.W. Schlag and M. Quack (Eds.): “Intermolecular Processes”, Ber. Bunsenges. Phys. Chem. 92 (1988) 3; J. Manz, C.S. Parmenter, R.M. Hochstrasser and G.L. Hofacker (Eds.): “Mode Selectivity in Unimolecular Reactions”, Chem. Phys. 139 (1989) 1.Google Scholar
  11. [11]
    S.K. Gray, S.A Rice and D.W. Noid: J. Chem. Phys. 84 (1986) 3745; R.T. Skodje, M.J. Davis: J. Chem. Phys. 88 (1988) 2429; M.J. Davis: J. Chem. Phys. 85 (1985) 1016; M.J. Davis: J. Chem. Phys. 86 (1987) 3978.ADSCrossRefGoogle Scholar
  12. [12]
    T. Uzer: Phys. Rep. 199 (1991) 75.ADSCrossRefGoogle Scholar
  13. [13]
    D.W. Oxtoby and S.A. Rice: J. Chem. Phys. 65 (1976) 1676.ADSCrossRefGoogle Scholar
  14. [14]
    C. Jaffe and P. Brumer: J. Chem. Phys. 73 (1980) 5646.MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    E.L. Sibert, W.P. Reinhardt and J.T. Hynes: J. Chem. Phys. 11 (1982) 3583.ADSCrossRefGoogle Scholar
  16. [16]
    C.C. Martens and G.S. Ezra: J. Chem. Phys. 87 (1987) 284; G.S. Ezra, C.C. Martens and L.E. Fried: J. Phys. Chem. 91 (1987) 3721; C.C. Martens, M.J. Davis and G.S. Ezra: Chem. Phys. Lett. 142 (1987) 519.ADSCrossRefGoogle Scholar
  17. [17]
    Y.M. Engel and R.D. Levine: Chem. Phys. Lett. 164 (1989) 270.ADSCrossRefGoogle Scholar
  18. [18]
    The use of action-angle variable also makes the correspondence with quantum mechanics particularly simple: The vibrational quantum numbers are the action variables measured units of h/2π. The zero order Hamiltonian H0(I) can thus be inferred from a Dunham-type fit of the energy as an analytic function of vibrational quantum numbers to the molecular spectrum. Of course, such a fit will be increasingly less perfect as the energy is increased, due to the breakdown of the zero order separable approximation.Google Scholar
  19. [19]
    D.E. Weeks and R.D. Levine: Phys. Letts. A, 167 (1992) 32.MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    G.E. Ewing: J.Phys. Chem. 91 (1987) 4662.CrossRefGoogle Scholar
  21. [21]
    J.A. Beswik and J. Jortner: Adv. Chem. Phys. 47 (1981) 363.CrossRefGoogle Scholar
  22. [22]
    Although typically not done, it is possible to employ standard higher order perturbative techniques to reintroduce the sub-resonant and super-resonant terms ignored by the simple process of averaging (see for example Ref. 4, p. 107). These terms then give rise to secondary islands, the largest of which correspond to the sub-resonant terms.Google Scholar
  23. [23]
    Loosely speaking, higher order resonances can ‘borrow width’ from lower order ones thereby increasing their effective coupling strength. The effective width of a higher order resonance is therefore increased by lower order resonances which ‘mix into’ the higher order one.Google Scholar
  24. [24]
    T.A. Holme and R.D. Levine: Chem. Phys. 131 (1989) 169.ADSCrossRefGoogle Scholar
  25. [25]
    I. Benjamin, O.S. van Roosmalen and R.D Levine: J. Chem. Phys. 81 (1984) 3352; O.S. van Roosmalen, I. Benjamin and R.D. Levine: J. Chem. Phys. 81 (1984) 5986.ADSCrossRefGoogle Scholar
  26. [26]
    G. Strey and I.M. Mills: J. Mol. Spec. 59 (1976) 103.ADSCrossRefGoogle Scholar
  27. [27]
    R.P. Muller, J.S. Hutchinson and TA. Holme: J. Chem. Phys. 90 (1989) 4582.ADSCrossRefGoogle Scholar
  28. [28]
    R. Fleming and J.S. Hutchinson: Comp. Phys. Comm. 51 (1988) 13.CrossRefGoogle Scholar
  29. [29]
    M. Iwai and R.D. Levine: Phys. Rev. A42 (1990) 3991.ADSGoogle Scholar
  30. [30]
    Y.S. Li, R.M. Whitnell, K.R. Wilson and R.D. Levine, J. Phys. Chem. 91 (1993) 3647.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Fritz Haber Research Centre for Molecular DynamicsThe Hebrew UniversityJerusalemIsrael

Personalised recommendations