Michael E. Kellman

Algebraic resonance dynamics of the normal/local transition from experimental spectra of ABA triatomics

An algebraic transformation is used to demonstrate the exact equivalence of the local and normal mode Hamiltonians for coupled anharmonic stretches. This SU(2) model is then interpreted semiclassically to extract quantitative information about nonlinear resonances in ABA triatomics from the Darling–Dennison spectral fit. A ‘‘glossary’’ is presented which makes it very easy to translate between the SU(2) language and standard spectroscopic terminology. In spectra predicted from the Darling–Dennison fit, transitions from a normal to local mode level pattern in molecules such as O3 are easily interpreted semiclassically in terms of trajectories in action/angle space and dynamical barriers. Although the local and normal algebraic Hamiltonians are equivalent for spectral fitting, local modes have the desirable property that they admit a simple representation in the coordinate picture. Local modes such as Morse oscillators therefore are the preferred physical starting point for stretching vibrations of general ...

Unified semiclassical dynamics for molecular resonance spectra

A method is presented to depict the intramolecular dynamics of resonantly coupled vibrations, starting from the experimental overtone and combination spectrum. The nonlinear least‐squares 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 semic...

Phase space bifurcation structure and the generalized local‐to‐normal transition in resonantly coupled vibrations

The 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 order. 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 e...

Fermi resonance phase space structure from experimental spectra

The method of algebraic resonance dynamics is developed to extract semiclassical dynamics of molecules from the fit to the experimental spectrum of bend and stretch normal modes with 2:1 Fermi resonance. The Heisenberg correspondence principle is used to determine phase space profiles for resonance polyads. Each quantum level in a polyad corresponds to a trajectory in action‐angle phase space. Profiles for experimental spectra of CO2 and a series of substituted methanes show a common phase space structure with four distinct types of semiclassical motion.

Semiclassical phase space evolution of Fermi resonance spectra

The evolution of the semiclassical phase space of a Fermi resonance spectrum is investigated as the strength of the resonance coupling is varied between zero and the strong coupling limit. The phase space evolution gives information beyond that contained in the phase space profile of the experimental spectrum alone. The zero‐order phase space is found to be different in important respects from that of the pendulum model of a nonlinear resonance. In the weak coupling regime, the phase space evolution is essentially like that of a dynamical barrier picture. In the strong coupling regime of ‘‘intrinsic resonance,’’ the phase space structure is much different. Topology change appears to take place in a more discontinuous manner than in the weak coupling regime. The phase space evolution shows that some levels are problematic for an adiabatic switching treatment. The origin of some anomalous levels seen both in phase space profiles of experimental spectra and in semiclassical quantization studies is clarified.

New raising and lowering operators for highly anharmonic coupled oscillators

New raising and lowering operators A+, A are proposed for vibrations in the highly anharmonic regime (roughly corresponding to the zone of overlapping nonlinear resonances and the transition to classical chaos). All matrix elements 〈ν+k‖0‖ν〉 of p and powers xλ of x=2Ce−aq between bound states ‖ν〉 of the Morse oscillator can be written exactly in terms of powers of the new operators, (A+)k and (A)k, and a supplementary operator α. In terms of α, A+, A, the momentum and coordinate operators take a form similar to that of the harmonic oscillator in terms of a+, a. It is shown that it is the operator α that is crucial for representing the novel qualitative behavior of dynamical operators in the highly anharmonic regime. It is therefore suggested that the operators α(A+)k and α(A)k be used in place of the conventional (a+)k and (a)k in fits and dynamical models for systems of coupled anharmonic vibrations.

Algebraic representation for multiple anharmonic oscillators via compact and non-compact unitary groups

Abstract Coupling of two and three Morse oscillators is treated algebraically using the compact and non-compact Lie groups SU( n ) and SU( n , 1). A model hamiltonian which separates into harmonic, anharmonic, and coupling terms is expressed simply in terms of the group generators.

Vibrational dynamics of ABA triatomics in a semiclassical algebraic approach

Abstract An algebraic formalism allows one to transform freely between complementary zero-order local and normal representations of vibrational dynamics of ABA triatomics. A semiclassical Interpretation of the local mode representation recovers the picture of Sibert et al. of an effective rotating top hindered by a double-well barrier. The complementary, but previously elusive normal representation pictures the vibrations as a top in an effective magnetic field. Darling—Dennison normal mode coupling due to off-diagonal unharmonic effects is represented by a quadruple-well barrier. In this normal representation, states with normal character are above the dynamical barrier, and states with local character are trapped.

Algebraic resonance quantization of coupled anharmonic oscillators

A method called algebraic resonance quantization (ARQ) is presented for highly excited multidimensional systems. This approach, based on the Heisenberg form of the correspondence principle, is a fully quantum mechanical matrix method. At the same time, it uses modern nonlinear classical mechanics to greatly simplify the Hamiltonian matrix. For a model system of coupled Morse oscillators, a nonlinear resonance analysis shows that the Hamiltonian matrix is dominated by a few leading terms. This leads to an effective truncated sparse matrix whose diagonalization yields eigenvalues in excellent agreement with the exact values, even high in the chaotic regime. A new finding is that quantum couplings corresponding to rapidly oscillating, nonresonant terms can be important, and not just the higher‐order resonant terms. The generalization of ARQ via a numerical semiclassical technique to many‐dimensional systems with arbitrary couplings is outlined. The applicability of contemporary vector methods from quantum chemistry to ARQ of high vibrational levels is considered. The feasibility of sparse matrix ARQ methods for fitting spectra in the highly chaotic, ‘‘unassignable’’ regime is discussed.

New assignment of vibrational spectra of molecules with the normal-to-local modes transition

Abstract A new assignment is proposed for vibrational spectra of molecules with the normal-to-local modes transition. Using information contained on the polyad phase sphere, levels with local mode dynamics are assigned with local mode quantum numbers, and levels with normal mode dynamics are assigned with normal mode quantum numbers.