William P. Reinhardt

Intramolecular vibrational relaxation and spectra of CH and CD overtones in benzene and perdeuterobenzene

A theoretical model is presented for the vibrational dynamics of highly excited CH and CD overtones in benzene and perdeuterobenzene. The origin, path, and time scale for the overtone relaxation are described. The critical near resonant interaction responsible for the energy flow from an excited CH(D) oscillator to the ring is a Fermi resonance coupling, identified by Sibert, Reinhardt, and Hynes [Chem. Phys. Lett. 92, 455 (1982)]. Quantum overtone spectra are calculated both from time independent and time dependent perspectives and good qualitative agreement is found with the experimental overtone spectra of Reddy, Heller, and Berry [J. Chem. Phys. 76, 2814 (1982)]. Some expected consequences for future experiments on benzene and related systems are indicated.

Adiabatic separations of stretching and bending vibrations: Application to H2O

A detailed investigation is made into the use of adiabatic approximations for describing excited stretching and bending vibrations of the water molecule. The goal is to determine precisely how effective this approach can be in a fully quantum mechanical triatomic calculation which incorporates anharmonicities to all orders in each of the modes. Great care is taken to avoid introducing unnecessary limitations or approximations: (i) Curvilinear coordinates are used rather than the Cartesian coordinates which form the starting point for normal mode calculations; (ii) the exact quantum kinetic energy operator in these coordinates is used as the basis for both the adiabatic and full three‐dimensional calculations; (iii) a Sorbie–Murrell‐type potential energy surface is used, giving a reasonable representation of the ground electronic surface for large excursions from the equilibrium configuration. In addition to the bond and bond‐angle variables of earlier local mode investigations, a slightly different set of...

Approximate constants of motion for classically chaotic vibrational dynamics: Vague tori, semiclassical quantization, and classical intramolecular energy flow

Coupled nonlinear Hamiltonian systems are known to exhibit regular (quasiperiodic) and chaotic classical motions. In this and the preceding paper by Jaffe and Reinhardt, we find substantial short time regularity even in the chaotic regions of phase space for what appears to be a large class of systems. This regularity is demonstrated by the behavior of approximate constants of motion calculated by Pade summation of the Birkhoff–Gustavson normal form expansion and is attributed to remnants of destroyed invariant tori in phase space. The remnant toruslike manifold structures are used to suggest justification for use of Einstein–Brillouin–Keller semiclassical quantization procedures for obtaining quantum energy levels even in the absence of complete tori and to form a theoretical basis for the calculation of rate constants for intramolecular mode–mode energy transfer. These results are illustrated in a thorough analysis of the Henon–Heiles oscillator problem. Possible generality of the analysis is shown by b...

Classical dynamics of energy transfer between bonds in ABA triatomics

A discussion of energy transfer betwen bonds is presented for ABA triatomics for which a local mode picture affords a good zeroth‐order description. A model of H2O is used to illustrate the discussion. It is shown that the energy transfer between the bonds can be readily understood both qualitatively and quantitatively in terms of the dynamics of a twofold hindered rotor. The transformation to this representation leads to a simple picture of the transition from ’’local mode’’ to ’’normal mode’’ behavior and to simple analytic expressions for the energy transfer rates. These predicted rates are found to be in excellent agreement with trajectory calculations. Isotope effects are discussed for the H2O–D2O pair.

Classical dynamics of highly excited CH and CD overtones in benzene and perdeuterobenzene

The classical decay of highly excited CH and CD overtones of benzene and perdeuterobenzene is examined by trajectory techniques. The character, mechanism, and time scale of these decays are in striking qualitative agreement with those found in the preceding quantum study by Sibert, Reinhardt, and Hynes. The classical coupling mechanism for the decay is shown to be nonlinear Fermi resonance involving the CH(D) stretch and the in‐plane CCH(D) wag. The decay path is described in terms of overlapping nonlinear resonance interactions.

Quantum mechanics of local mode ABA triatomic molecules

The quantum mechanics of the vibrational stretching dynamics in ’’local mode’’ triatomic molecules is examined. A model for H2O is taken as a prototype. The quantum analysis exploits the corresponding classical analysis of the companion paper, in which an approximate but accurate Hamiltonian is derived via the techniques of nonlinear mechanics. Quantization of this Hamiltonian gives H2O vibrational energies in excellent agreement with direct quantum calculations. The corresponding overtone‐combination spectrum of the H2O model is analyzed in terms of local and normal mode behavior with the aid of the twofold hindered rotor perspective provided by the Hamiltonian. The splittings in the spectrum are related to the quantum dynamics of energy transfer. A semiclassical WKB analysis is also used to relate the splittings to classical energy transfer rates and quantum dynamical tunneling and reflection probabilities.

Uniform semiclassical quantization of regular and chaotic classical dynamics on the Hénon–Heiles surface

Qualitative arguments (made substantially more quantitative in the accompanying article by Shirts and Reinhardt) are put forward which indicate that the apparently chaotic dynamics on the Henon–Heiles surface display sufficient regularity on a short to intermediate (but not long) time scale to allow use of standard EBK quantization techniques, taking advantage of the remnants of manifold structure that these remarks imply. A complete uniform semiclassical quantization is carried out using the time independent technique of the Birkhoff–Gustavson normal form, recently introduced in the context of semiclassical quantization by Swimm and Delos.

Nonlinear resonances and vibrational energy flow in model hydrocarbon chains

Intramolecular energy transfer in hydrocarbons is modeled with an anharmonic HC bond coupled to a chain of harmonically coupled carbon atoms. The HC stretch is initially excited to various vibrational states and the flow of energy from the HC bond is observed. It is found that, at sufficiently high excitation, vibrational energy flow is irreversible and occurs on a time scale of roughly 100 fs, independent of chain length. This corresponds very well to experimentally observed spectral bandwidths. By analyzing the motion in a set of natural dynamical modes, it is shown that this energy transfer is due to sequential nonlinear resonances in the chain. Analytic modeling of the mode–mode couplings then provides a simple approximate description of the energy transfer process.

The semiclassical quantization of nonseparable systems using the method of adiabatic switching

A method for the semiclassical quantization of multidimensional bound systems based on the adiabatic hypothesis is examined. The validity criteria for multidimensional adiabaticity is discussed. It is demonstrated that the quantizing orbits for nonseparable systems can often be obtained by propagating a single trajectory from well defined initial conditions with a time‐dependent Hamiltonian for ∼100 periods. Numerical examples using systems with up to five degrees of freedom are presented and show generally excellent results. It is shown that this method can be used to quantize some states using chaotic trajectories.

Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity

In this second of two papers, we present all stationary so- lutions of the nonlinear Schrodinger equation with box or pe- riodic boundary conditions for the case of attractive nonlin- earity. The companion paper has treated the case of repulsive nonlinearity. Our solutions take the form of stationary trains of bright solitons. Under box boundary conditions the solu- tions are the bounded analog of bright solitons on the innite line, and are in one-to-one correspondence with particle-in-a- box solutions to the linear Schrodinger equation. Under peri- odic boundary conditions we nd several classes of solutions: constant amplitude solutions corresponding to boosts of the condensate; the nonlinear version of the well-known particle- on-a-ring solutions in linear quantum mechanics; nodeless, real solutions; and a novel class of intrinsically complex, node- less solutions. The set of such solutions on the ring are de- scribed by the Cn character tables from the theory of point groups. We make experimental predictions about the form of the ground state and modulational instability. We show that, though this is the analog of some of the simplest problems in linear quantum mechanics, nonlinearity introduces new and surprising phenomena in the stationary one-dimensional non- linear Schrodinger equation. We also note that in various limits the spectrum of the nonlinear Schrodinger equation re- duces to that of the box, the Rydberg, and the harmonic oscillator, the latter being for repulsive nonlinearity, thus in- cluding the three most common and important cases of linear quantum mechanics.