Chaotic scattering on C4v four-disk billiards: Semiclassical and exact quantum theories.

The principal families of periodic orbits that emerge from the stationary points of the six-dimensional potential energy surface of the C2H2 molecular system, as well as periodic orbits from saddle-node bifurcations, have been located and propagated for an energy range up to 36 500 cm−1 above the absolute minimum of the potential. The bifurcation diagrams of these periodic orbits reveal the regions of phase space where the dynamics are regular or chaotic (with soft or hard chaos) for acetylene, vinylidene, and the region over these two isomers. An association of the structure of phase space with spectroscopic findings is made by calculating Gutzwiller’s semiclassical trace formula and classical survival probability functions.

We show that the analytic multidimensional semiclassical tunneling formula of Miller et al. [Miller, W. H.; Hernandez, R.; Handy, N. C.; Jayatilaka, D.; Willets, A. Chem. Phys. Lett. 1990, 172, 62] is qualitatively incorrect for deep tunneling at energies well below the top of the barrier. The origin of this deficiency is that the formula uses an effective barrier weakly related to the true energetics but correctly adjusted to reproduce the harmonic description and anharmonic corrections of the reaction path at the saddle point as determined by second order vibrational perturbation theory. We present an analytic improved semiclassical formula that correctly includes energetic information and allows a qualitatively correct representation of deep tunneling. This is done by constructing a three segment composite Eckart potential that is continuous everywhere in both value and derivative. This composite potential has an analytic barrier penetration integral from which the semiclassical action can be derived and then used to define the semiclassical tunneling probability. The middle segment of the composite potential by itself is superior to the original formula of Miller et al. because it incorporates the asymmetry of the reaction barrier produced by the known reaction exoergicity. Comparison of the semiclassical and exact quantum tunneling probability for the pure Eckart potential suggests a simple threshold multiplicative factor to the improved formula to account for quantum effects very near threshold not represented by semiclassical theory. The deep tunneling limitations of the original formula are echoed in semiclassical high-energy descriptions of bound vibrational states perpendicular to the reaction path at the saddle point. However, typically ab initio energetic information is not available to correct it. The Supporting Information contains a Fortran code, test input, and test output that implements the improved semiclassical tunneling formula.

Semiclassical quantum unimolecular reaction rate theory revisited

We first give a short outline of the semiclassical periodic orbit theory that allows one to relate quantum shell fluctuations in a mean-field system to the periodic orbits of the corresponding classical system. Earlier applications to gross-shell effects in nuclei (ground-state deformations), metal clusters (ground-state deformations and supershells), and quantum dots (conductance oscillations in weak magnetic fields) are briefly summarized. We then report on a recent simple and transparent semiclassical interpretation of the origin of the mass asymmetry in the fission of heavy nuclei. Using only a few classical periodic orbits and a cavity model for the nuclear mean field, we explain the onset of left-right asymmetric shapes at the fission isomer minimum by the principle of constant action of the shortest orbits.

Using the classical (CSC), primitive (PSC), and uniform (USC) semiclassical expressions for transition probabilities given by Miller and co-workers, we have calculated the reactive and nonreactive 0 → 0 and 0 → 1 transition probabilities for the collinear H + H2 exchange reaction. Comparison with previously calculated exact quantum and quasiclassical results for the reactive and nonreactive 0 → 0 transitions reveals that the semiclassical approximations are not very good, especially the CSC and PSC ones. All three semiclassical probabilities for the reactive 0 → 0 transition exceed unity in the collision energy range from 0.0 to 0.2 eV above the quasiclassical reaction threshold. This feature coupled with the failure of any of the semiclassical approximations to produce the marked quantum effects present in this transition causes these results to be less accurate than the corresponding quasiclassical ones. For the reactive and nonreactive 0 → 1 transitions the USC results are in qualitative agreement with the exact quantum ones and are better than the standard quasiclassical results. However, the reverse quasiclassical results are almost as good as the USC ones for these transitions. A probable reason for the inability of the USC expression to produce the strong oscillations observed in the exact quantum results is that the latter are due to interference between direct and resonant (i.e., compound state) processes whereas the present formulation of the semiclassical method does not encompass such phenomena. A comparison of the total reaction probabilities obtained by the USC and quasiclassical methods with the exact quantum one indicates that the USC result is more accurate than the quasiclassical one, except at collision energies less than 0.50 eV. This improved accuracy is due to a partial cancellation of errors in the contributing 0 → 0 and 0 → 1 USC reactive transition probabilities.

Classical and quantum mechanics of a strongly chaotic billiard system

We perform an analytic semi-classical quantization of the straight QCD string with one end fixed and a massless quark on the other, in the limits of orbital and radial dominant motion. Our results well approximate those of the exact numerical semi-classical quantization as well as our exact numerical canonical quantization.

We investigate theories in which classical and quantum-mechanical degrees of freedom interact dynamically. In commonly used semiclassical theories, such as those used to study inflationary-universe models, quantum fluctuations do not affect the dynamics of the classical variables. We construct a new semiclassical theory in which the quantum and classical fluctuations do affect each other; the Wigner probability function turns out to be a special case. Relevance to calculations of perturbations from inflation are discussed.

The Weyl representation in classical and quantum mechanics

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.

Eigenlevel correlation diagrams has proven to be a very useful tool to understand eigenstate characteristics of classically chaotic systems. In particular, we showed in a previous publication [Phys. Rev. Lett. 80, 944 (1998)0031-900710.1103/PhysRevLett.80.944] how to unveil the scarring mechanism, a cornerstone in the theory of quantum chaos, using the Planck constant as the correlation parameter. By increasing the Planck constant, we induced a transition from order to chaos, in which scarred wave functions appeared as the interaction of pairs of eigenstates in broad avoided crossings, forming a well-defined frontier in the correlation diagram. In this paper, we demonstrate that this frontier can be obtained by means of the semiclassical quantization of the involved scarring periodic orbits. Additionally, in order to calculate the Maslov index of each scarring periodic orbit, which is necessary for the semiclassical quantization procedure, we introduce a straightforward method based on Lagrangian descriptors. We illustrate the theory using the vibrational eigenstates of the LiCN molecular system.

Time Evolution in an External Field: The Unitarity Paradox

Semiclassical theory of hyperspherical vibrational resonances in ABA molecules

We present a semiclassical quantization condition, i.e., quantum-classical correspondence, for steady states of nonadiabatic systems consisting of fast and slow degrees of freedom (DOFs) by extending Gutzwiller's trace formula to a nonadiabatic form. The quantum-classical correspondence indicates that a set of primitive hopping periodic orbits, which are invariant under time evolution in the phase space of the slow DOF, should be quantized. The semiclassical quantization is then applied to a simple nonadiabatic model and accurately reproduces exact quantum energy levels. In addition to the semiclassical quantization condition, we also discuss chaotic dynamics involved in the classical limit of nonadiabatic dynamics.

The following one-dimensional model potential barriers are studied to compute the tunneling probabilities using semiclassical transition state theory (SCTST) based on fourth order vibrational perturbation theory (VPT4): (i) Morse barrier, (ii) asymmetric Eckart barrier, (iii) adjacent well barrier, and (iv) first order approximation to the symmetric Eckart barrier. The current investigation extends a previous study performed for the symmetric Eckart barrier. The performance of VPT4-SCTST is investigated for a broad range of energies, including the very deep tunneling regime (approximately 1% of the barrier height), and the results are compared to the exact quantum mechanical treatment, as well as to the popular second order vibrational perturbation theory (VPT2)-SCTST approach. While the VPT4-SCTST results are in excellent agreement with their quantum mechanical counterparts for a wide range of energies above and below the barrier, it is found for systems (ii) and (iv) that higher order terms (in ) in energy expansion are important for similarly excellent agreement in the deep tunneling regime. For the asymmetric Eckart barrier, the convergence properties of SCTST are analyzed in terms of its analytically known Siegert eigenvalues. A simple empirical modification of VPT2-SCTST advocated in the aforementioned previous work for the symmetric Eckart barrier is also tested here for the new model systems, and it is shown not to provide a uniformly systematic improvement.