At present, two main types of algebraic methods are employed for analysis of molecular spectra. The first goes back to the early days of molecular spectroscopy. The second, developed recently by Iachello and coworkers, grew out of nuclear physics and makes use of classical Lie algebras such as SU(4). In this review, the standard spectroscopic fitting Hamiltonian for molecular vibrations, including resonance interactions, is first described. Then, new developments in the application of the standard approach are surveyed. In particular, the question of how one determines the true nature of molecular motions in highly excited spectra is investigated. Next, the recent algebraic approach of Iachello and coworkers is discussed. Application of ideas of molecule-like modes and algebraic methods to the analysis of the electronic spectra of atoms is discussed. Finally, prospects for future development of algebraic methods are discussed.

Phase space bifurcation structure and the generalized localtonormal transition in resonantly coupled vibrationsCatastrophe theory is used to classify the dynamics of spectra of resonantly coupled vibrations, based on earlier work on the bifurcation structure of the Darling-Dennison and 2: I Fermi resonance fitting Hamiltonians. The goal is a generalization of the language of the "normallocal transition" to analyze experimental spectra of general resonant systems. The set of all fixed points of the Hamiltonian on the polyad phase sphere for all possible molecular parameters constitutes the catastrophe manifold. The projection of this manifold onto the subspace of molecular parameters is the catastrophe map. The map is divided into zones; each zone has its own characteristic phase sphere structure. The taxonomy of global phase sphere structures within all zones gives the classification of the semiclassical dynamics. The 1: I system, with normal-local transition, is characterized by cusp catastrophes, with elementary pitchfork bifurcations. In contrast, the 2: 1 system is characterized by fold catastrophes, with elementary transcritical bifurcations. The catastrophe map can be used in a new method to classify experimental spectra on the basis of the system's underlying semiclassical dynamics. The catastrophe map classification appears to persist for nonintegrable, chaotic Hamiltonians, indicating the utility of catastrophe theory for understanding the morphology of chaotic systems.

Articles you may be interested in Polyad quantum numbers and multiple resonances in anharmonic vibrational studies of polyatomic moleculesA theory of approximate dynamical constants of motion is presented for vibrational (and implicitly, rovibrational) spectra of poly atomics with multiple nonlinear resonances. The formalism is developed in terms of simple vector algebra. The theory is applied to Hamiltonians used in fits of experimental spectra of H 2 0, CHCIF 2 , and acetylene, with attention to reduced dimension motion, assignability of spectra, and statistical analysis of chaotic spectra. The approximate constants may be of interest as bottlenecks to vibrational energy flow in polyatomics. They may also be useful in reducing the size of basis sets in quantum-mechanical calculations of rotation-vibration spectra.

A method is presented to depict the intramolecular dynamics of resonantly coupled vibrations, starting from the experimental overtone and combination spectrum. The nonlinear leastsquares fit of the spectrum is used to obtain a semiclassical phase space Hamiltonian via the Heisenberg correspondence principle. This integrable Hamiltonian, corresponding to quasiperiodic motion, is used to generate a classical trajectory in phase space for each energy level in a resonance polyad. Polyad phase space profiles are shown to have complete mutual consistency starting from a fit in either the local or normal representation. It is argued that the best way to depict the phase space profile is on a spherical surface called the polyad phase sphere. Represented in this way, the local and normal mode phase spaces are seen to be a single entity, manifestly equivalent by a 90° rotation. The phase space trajectories can be converted into a coordinate space representation. This gives an easily visualized picture of the semiclassical intramolecular dynamics corresponding to each energy level. The polyad phase spheres from the fits of the experimental stretching spectra of H 2 0, 0 3 and S02 are displayed. H 2 0 and 0 3 are seen to be molecules with a local to normal modes transition, while S02 is seen to be very near the pure normal modes limit. The experimentally determined phase space dynamics of H 2 0 seen on the phase sphere are compared with the dynamics determined by Lawton and Child from trajectory calculations on the Sorbie-Murrell potential surface. The coordinate space trajectories corresponding to the phase spheres are compared with wave functions from the fit of the spectrum.

The vibrational spectrum of HCP ͑phosphaethyne͒ is studied and analyzed in terms of a 1:2 resonance effective Hamiltonian. The parameters of the model Hamiltonian are determined by fitting 361 out of the first 370 energy levels obtained from diagonalization of the full Hamiltonian, which is based on a newly calculated potential-energy surface with near spectroscopic accuracy. It is demonstrated that all features characteristic of the approach to the HCP↔CPH isomerization, such as the strong mixing between the bending and CP-stretching motions, the appearance of ''isomerization states'' ͑large amplitude bending motion͒ at intermediate energies, and the diagnostically significant appearance of a zigzag pattern in the energy spacings between neighboring levels within each polyad, are quantitatively reproduced by the effective Hamiltonian. The semiclassical analysis of the model Hamiltonian for specific combinations of the HC-stretch and polyad quantum numbers explains all of the observed features of the full Hamiltonian in terms of stable and unstable periodic orbits. In particular, the birth of the isomerization states is found to be related to a saddle-node bifurcation of the classical phase space. The connection with the ''polyad phase sphere'' representation of quantum polyads is also discussed.

The normal to local mode transition in AB2 triatomic molecules: The susceptibility of eigenstates to symmetry breaking perturbations

Articles you may be interested inA study on bifurcations and structure of phase space concerning intrinsic localized modes in a nonlinear magneto-mechanical lattice AIP Conf.Isotope effect in normal-to-local transition of acetylene bending modes Catastrophe map classification of the generalized normal-local transition in Fermi resonance spectraThe generalization of the local-to-normal transition seen in symmetric triatomics is considered for nonsymmetric molecules and 2: 1 Fermi resonance systems. A straightforward generalization based on a division of phase space into local and normal regions is not possible. Instead, classification of the phase space bifurcation structure is presented as the complete generalization of the local-normal concept for all spectroscopically relevant systems of two vibrations interacting via a single nonlinear resonance. The polyad phase sphere (PPS) is shown to be the natural arena to analyze the bifurcation structure for resonances of arbitrary o:der. For 1: 1 and 2: 1 resonances, the bifurcation problem is reduced to one or two great Circles on the phase sphere. All bifurcations are shown to be examples of elementary bifurcations of vector fields in one dimension. The classification of the bifurcation structure is therefore governed and greatly simplified by the theory of the universal unfolding and codimension of elementary bifurcations. The implications for large-scale bifurcation structure and transport in molecules with chaotic motion are briefly discussed. a) Address after December 15, 1989:

Approximate constants of motion of acetylene (C2H2), analyzed previously below 10 000 cm−1, are determined from analysis of a nonlinear least squares fit of the highly excited vibrational absorption spectrum. Although there are at least ten distinct Fermi resonance couplings in the measured spectrum up to 24 000 cm−1, there is one, and quite possibly two, good constants of motion. These constants are pointed out to be equivalent to a preferred energy transfer pathway discussed by Smith and Winn. It is suggested that these constants may also apply to ‘‘unassignable’’ stimulated emission pumping spectra, which sample a different region of phase space.