Gregory S. Ezra

Symmetry properties of molecules

1 The molecular Hamiltonian.- 1.1 The molecular kinetic energy.- 1.2 Quasi-rigid molecules.- 1.3 The Wilson-Howard-Watson Hamiltonian.- 1.4 Nonrigid molecules.- 1.5 Other approaches to nonrigid molecule dynamics.- 2 Symmetry properties of rigid molecules.- 2.1 Fundamental symmetries of the molecular Hamiltonian.- 2.2 Symmetry properties of rigid molecules - Formalism.- 2.3 Symmetry properties of rigid molecules - Interpretation.- 2.4 Symmetry properties of diatomic molecules.- 3 Symmetry properties of nonrigid molecules.- 3.1 The symmetry group of the semi-rigid molecular model..- 3.2 The structure of H..- 3.3 Isodynamic operations.- 3.4 The isometric theory.- 4 Nonrigid molecule symmetry groups.- 4.1 The XY3 invertor.- 4.2 Coaxial rotors with non-identical ends.- 4.3 Coaxial rotors with identical ends.- 4.4 Some other rotors.- 4.5 Pseudorotation of ZXY4.- 4.6 Conclusions and possibilities for future develoments.- Appendix 1 Rotations of axes and the group 0(3).- Appendix 2 Angular momentum and rotational wavefunctions.- A2.1 Angular momentum operators.- A2.2 The symmetry group of the spherical rotor.- A2.3 Construction of rotational wavefunctions.- Appendix 3 The class structure and irreducible representations of semi-direct products.- A3.1 Vocabulary.- A3.2 The class structure of G K.- A3.3 The irreducible representations of G K.- Appendix 4 The point groups as semi-direct products.- A4.1 Cyclic and dihedral groups.- A4.2 Cubic groups.- A4.3 Icosahedral groups.- References.

Semiclassical quantization using classical perturbation theory: Algebraic quantization of multidimensional systems

The method of algebraic quantization, a semiclassical analog of Van Vleck perturbation theory, is applied to multidimensional resonant, nonresonant, and nearly resonant systems. perturb, a special purpose program written in C, is utilized to implement classical perturbation theory efficiently to high order. States corresponding to both regular and chaotic classical regimes are quantized, and accurate eigenvalues obtained in both cases. Various quantization rules are compared, and a novel symmetry preserving rule is given which leads to good agreement with quantum mechanics. The method is able to reproduce purely quantum mechanical splittings to very good accuracy. Algebraic quantization combined with Pade resummation is used to determine energy eigenvalues for a resonant system with five degrees of freedom.

EBK quantization of nonseparable systems: A Fourier transform method

We present an accurate and efficient method for EBK quantization of bound states in multidimensional systems, based upon a Fourier representation of invariant tori in the quasiperiodic regime. The key feature of the method is the direct determination of good action variables from Fourier transforms of trajectories, using a formula due to Percival. It is possible to calculate actions using this formula provided frequencies and Fourier components can be determined to high accuracy. We describe a special windowing technique for this purpose, which overcomes the stability problems encountered in the recent related work of Eaker, Schatz, De Leon and Heller. Quantized energy levels can be found either by direct search for quantizing trajectories or by linear extrapolation from nearby arbitrary tori. Results are presented for two‐, three‐, and four‐dimensional vibrational Hamiltonians.

Classical and semiclassical mechanics of strongly resonant systems: A Fourier transform approach

The Fourier transform approach to EBK quantization, previously applied to nonresonant systems with up to four degrees of freedom [J. Chem. Phys. 83, 2990 (1985)], is extended to the case of strongly resonant classical motion. The classical mechanics of systems with 3:4, 1:2, and 1:1 resonances is examined in detail from the Fourier transform point of view, and the results of nonlinear resonance analysis used to interpret numerical trajectory Fourier spectra. Calculation of classical actions and numerical construction of the angle parametrization of invariant tori is described, and the relation between spectral frequency assignments and the choice of good action‐angle variables investigated. It is shown that correct quantization conditions for arbitrary resonant motion can be determined by direct numerical evaluation of Maslov indices. Semiclassical eigenvalues are reported for the 3:4, 1:2, and 1:1 resonant systems.

Nonadiabatic interactions in the photodissociation of ICN

Photodissociation of ICN in the A continuum has been modeled using classical trajectories assuming that all absorption from the linear ground state is to a single linear diabatic excited state which dissociates to form I*(2P1/2) and CN(2Σ+,v=0). It is also assumed that, in nonlinear excited state configurations, nonadiabatic transitions occur to a bent surface which correlates diabatically to ground state I(2P3/2) and CN(2Σ+,v=0). Empirical potential surfaces with frozen CN bond lengths are employed, while transitions between the surfaces are treated using either the Miller–Meyer classical electron model or a simple diabatic version of the Tully–Preston surface‐hopping model. With the above assumptions, the Miller–Meyer method is found to give much better agreement with the experimental results. Theoretical results obtained with the Miller–Meyer method are compared with recent experimental data on the I*/I branching ratio, the average CN rotational energies, and the product CN rotational distributions as...

Local frequency analysis of chaotic motion in multidimensional systems: energy transport and bottlenecks in planar OCS

The method of local frequency analysis for study of chaotic motion in multidimensional systems is described, and applied to the analysis of intramolecular energy flow in a three-mode model for OCS. The nature of possible partial barriers responsible for long-time correlations in multimode systems is discussed.

Transport and turnstiles in multidimensional Hamiltonian mappings for unimolecular fragmentation: Application to van der Waals predissociation

A four‐dimensional symplectic (Hamiltonian) mapping of the type studied by Gaspard and Rice is used to model the predissociation of the van der Waals complex He–I2. Phase space structure and unimolecular decay in this mapping are analyzed in terms of a general approach recently developed by Wiggins. The two‐dimensional area preserving map obtained by restricting the 4D map to the T‐shaped subspace is studied first. Both the Davis–Gray theory and the analog of the alternative RRKM theory of Gray, Rice, and Davis for discrete maps are applied to estimate short‐time decay rates. A four‐state Markov model involving three intramolecular bottlenecks (cantori) is found to give a very accurate description of decay in the 2D map at short to medium times. The simplest version of the statistical Davis–Gray theory, in which only a single intermolecular dividing surface is considered, is then generalized to calculate the fragmentation rate in the full 4D map as the ratio of the volume of a four‐dimensional turnstile l...

Semiclassical cycle expansion for the helium atom

We analyse the classical dynamics of near-collinear electron configurations in helium. The dynamics turns out to be fully chaotic. An application of periodic orbit quantization techniques yields the energy of doubly excited states with high accuracy. The analysis shows that near-collinear intra-shell resonances are associated with an asymmetric stretch like molion of the electron pair rather than the symmetric stretch motion along the Wannier ridge. The purpose of this letter is threefold. We first analyse the classical dynamics for near-collinear arrangements of the two electrons in the helium atom. We find strong evidence that the resulting motion is fully chaotic in the corresponding symmetry plane whereas the linearized motion of the plane is stable. We then apply modern semi- classical techniques to quantize the chaotic dynamics and obtain the energies of certain doubly excited states. Finally, our results (together with numerically highly accurate quantum mechanical calculations) unambiguously show that the widely accepted viewpoint of electron pair propagation along the Wannier ridge for doubly excited 11 Prrmanent address: Max-Planck lnrlitut Tir Kcmphyrik. Postfach 103980. 6900 Heidelberg. Federal Republic of Gemany.

EIGENSTATE ASSIGNMENTS AND THE QUANTUM-CLASSICAL CORRESPONDENCE FOR HIGHLY-EXCITED VIBRATIONAL STATES OF THE BAGGOT H2O HAMILTONIAN

In this paper we study the classical and quantum mechanics of the 3-mode Baggot vibrational Hamiltonian for H2O. Our aim is to classify and assign highly-excited quantum states based upon a knowledge of the classical phase space structure. In particular, we employ a classical template formed by the primary resonance channels in action space, as determined by Chirikov resonance analysis. More detailed analysis determining the exact periodic orbits and their bifurcations and families of resonant 2-tori for the Baggot Hamiltonian confirms the essential correctness of the Chirikov picture. It is emphasized that the primary periodic orbits alone do not define a suitable phase space skeleton; it is important to consider higher dimensional invariant structures, such as 2-tori and 3-tori. Examining the manifold of quantum states for a given superpolyad number P=n1+n2+nb/2 reveals sequences of eigenstates that progress along the classical resonance zones. These sequences provide insight into the nature of strongly...

Interaction between bending vibrations and molecular rotation: a model study

The classical mechanics of the interaction between bending vibration and molecular rotation is studied for a simple rigid bender model. A one-dimensional representation of rotational motion enables surfaces of section to be constructed. Quasiperiodic, resonant and chaotic motions are identified, and related to characteristic patterns of vibration-rotation energy flow.