Phase space structure of triatomic molecules

The bifurcation structure is investigated for a Hamiltonian for the three coupled nonlinear vibrations of a highly excited triatomic molecule. The starting point is a quantum Hamiltonian used to fit experimental spectra. This Hamiltonian includes 1:1 Darling–Dennison resonance coupling between the stretches, and 2:1 Fermi resonance coupling between the stretches and bend. A classical Hamiltonian is obtained using the Heisenberg correspondence principle. Surfaces of section show a pronounced degree of chaos at high energies, with a mixture of chaotic and regular dynamics. The large-scale bifurcation structure is found semianalytically, without recourse to numerical solution of Hamilton’s equations, by taking advantage of the fact that the spectroscopic Hamiltonian has a conserved polyad quantum number, corresponding to an approximate constant of the motion of the molecule. Bifurcation diagrams are analyzed for a number of molecules including H2O, D2O, NO2, ClO2, O3, and H2S.

[1]  F. Verhulst,et al.  Asymptotic Integrability and Periodic Solutions of a Hamiltonian System in $1:2:2$-Resonance , 1984 .

[2]  G. S. Ezra,et al.  ASSIGNING VIBRATIONAL SPECTRA OF HIGHLY EXCITED MOLECULES : CLASSICAL AND QUANTUM VIBRATIONAL DYNAMICS OF THE H2O MOLECULE , 1996 .

[3]  G. Herzberg.  Infrared and raman spectra , 1964 .

[4]  J. L. Kinsey,et al.  Broad spectral features in the stimulated emission pumping spectrum of acetylene , 1988 .

[5]  W. Harter.  SU(2) coordinate geometry for semiclassical theory of rotors and oscillators , 1986 .

[6]  M. E. Kellman,et al.  Catastrophe map classification of the generalized normal–local transition in Fermi resonance spectra , 1990 .

[7]  Robert L. Weber,et al.  Sources of Quantum Mechanics , 1968 .

[8]  W. J. Lafferty,et al.  The high resolution infrared spectrum of the 2ν2 + ν3 and ν1 + ν2 + ν3 bands of 14N16O2 , 1977 .

[9]  L. Fried,et al.  Semiclassical quantization using classical perturbation theory: Algebraic quantization of multidimensional systems , 1987 .

[10]  G. S. Ezra.  Periodic orbit analysis of molecular vibrational spectra: Spectral patterns and dynamical bifurcations in Fermi resonant systems , 1996 .

[11]  N. Handy,et al.  Theoretical rotational–vibrational spectrum of H2S , 1989 .

[12]  Heller,et al.  Semiclassical dynamics of chaotic motion: Unexpected long-time accuracy. , 1991, Physical review letters.

[13]  K. Miyazaki,et al.  Microwave spectrum of chlorine dioxide in excited vibrational states , 1986 .

[14]  Robert W. Field,et al.  Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping , 1995 .

[15]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[16]  M. E. Kellman.  Algebraic resonance dynamics of the normal/local transition from experimental spectra of ABA triatomics , 1985 .

[17]  J. Baggott.  Normal modes and local modes in H2 X: beyond the x, K relations , 1988 .

[18]  C. Rankin,et al.  The Classical S-Matrix: Linear Reactive Collisions of H + Cl2 , 1971 .

[19]  J. Champion,et al.  Manifestation of bifurcations and diabolic points in molecular energy spectra , 1990 .

[20]  W. Heisenberg.  A quantum-theoretical reinterpretation of kinematic and mechanical relations , 1925 .

[21]  M. E. Kellman,et al.  Phase space bifurcation structure and the generalized local-to-normal transition in resonantly coupled vibrations , 1990 .

[22]  K. Lehmann.  On the relation of Child and Lawton’s harmonically coupled anharmonic–oscillator model and Darling–Dennison couplinga) , 1983 .

[23]  John P. Rose,et al.  Spectral patterns and dynamical bifurcation analysis of highly excited vibrational spectra , 1995 .

[24]  J. Rose,et al.  Assigning spectra of chaotic molecules with diabatic correlation diagrams , 1996 .

[25]  M. E. Kellman,et al.  Potential energy surfaces from highly excited spectra using the bootstrap fitting method: Two‐dimensional surfaces for water and ozone , 1991 .

[26]  A. Barbe,et al.  Microwave and infrared study of the ν2 state of 16O3 and identification of the (ν3 + ν2)−ν2 band lines at 10 μm , 1978 .

[27]  Allan N. Kaufman,et al.  Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories , 1979 .

[28]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[29]  R. Seydel.  From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[30]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[31]  Els van der Aa.  First-order resonances in three-degrees-of-freedom systems , 1983 .

[32]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[33]  R. Prosmiti,et al.  Periodic orbits, bifurcation diagrams and the spectroscopy of C2H2 system , 1995 .

[34]  A. G. Robiette,et al.  On the relationship of normal modes to local modes in molecular vibrations , 1985 .

[35]  M. E. Kellman,et al.  New assignment of Fermi resonance spectra , 1990 .

[36]  Joseph B. Keller,et al.  Corrected bohr-sommerfeld quantum conditions for nonseparable systems , 1958 .

[37]  QUALITATIVE ANALYSIS OF VIBRATIONAL POLYADS: N-MODE CASE , 1989 .

[38]  A. Barbe,et al.  Infrared spectra of 16O3 and 18O3: Darling and Dennison resonance and anharmonic potential function of ozone , 1974 .

[39]  R. Redding,et al.  The 4750 Å band system of chlorine dioxide , 1969 .

[40]  C. Jaffé.  Comment on: ‘‘Semiclassical phase space evolution of Fermi resonance spectra’’ , 1988 .

[41]  Eric J. Heller,et al.  Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits , 1984 .

[42]  Michael J. Davis.  Analysis of highly excited vibrational eigenstates , 1995 .

[43]  J. H. Van Vleck,et al.  The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics , 1928 .

[44]  M. E. Kellman,et al.  Unified semiclassical dynamics for molecular resonance spectra , 1989 .

[45]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[46]  L. L. Lohr,et al.  Ab initio synthesis of the ozone ultraviolet continuum , 1987 .

[47]  George Haller,et al.  Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems , 1996 .

[48]  David Potter.  Computational physics , 1973 .

[49]  M. E. Kellman.  Group theory of coupled oscillators: Normal modes as symmetry breaking , 1982 .

[50]  M. E. Kellman,et al.  ASSIGNING VIBRATIONAL SPECTRA OF CHAOTIC MOLECULES , 1995 .

[51]  Alan Weinstein,et al.  Normal modes for nonlinear hamiltonian systems , 1973 .

[52]  Calculation of linear stability boundaries for equilibria of Hamiltonian systems , 1987 .

[53]  Guozhen Wu.  The quasi-separable semiclassical dynamical subspace of strong Fermi resonance under SU(3) algebra , 1992 .

[54]  M. E. Kellman.  Algebraic methods in spectroscopy. , 1995, Annual review of physical chemistry.