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  • 1
    Electronic Resource
    Electronic Resource
    Classical and quantal pseudoergodic regions of the Henon–Heiles system (1988)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 88 (1988), S. 5688-5703 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: This work describes an approximate and temporary form of ergodicity, called pseudoergodicity. This is a classical phenomenon that is of particular interest since it has a straightforward quantum analog and can be expected to influence the dynamics of a quantum system more directly than rigorous ergodicity. A procedure is described for locating phase space regions where classical motion is pseudoergodic (pe) and this technique is applied to the Henon–Heiles system excited to near its escape energy. Certain pe zones containing chaotic trajectories appear to coincide with the vague tori of Reinhardt and co-workers while other such zones have less familiar forms. When the Henon–Heiles system is observed for long times, a single pe region becomes larger than others, thus marking incipient ergodicity on much of the energy shell. The classical pe regions are compared to quantum pe zones, calculated by a slight modification of a technique presented in an earlier paper. It is found that certain classical pe regions have close quantum analogs while others do not, for reasons probably related to the short time scale for quantum-classical correspondence. Among the classical regions that do not have quantum analogs is the aforementioned dominant pe zone. The implication is that the present quantum system does not display behavior similar to the truly large-scale ergodicity that occurs in the classical system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.454529
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  • 2
    Electronic Resource
    Electronic Resource
    Improved semiclassical propagation of wave packets (1989)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 91 (1989), S. 170-179 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The time-dependent variational method is applied to develop a potentially exact wave packet propagation technique that systematically improves the semiclassical Gaussian treatment of Heller. Difficulties encountered in previous applications of the variational method are avoided. Numerical examples are presented which demonstrate the accuracy and convergence of the present approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.457631
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  • 3
    Electronic Resource
    Electronic Resource
    The influence of classical resonances on quantum energy levels (1993)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 99 (1993), S. 3659-3668 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: In this paper, we examine the behavior of the quantum energy levels of a coupled oscillator system as the zero-order frequencies are varied to carry the corresponding classical system through a resonance. We find that the levels exhibit a pattern that is characteristic of the resonance. This pattern consists of clusters of levels, each containing a number of curves that run roughly parallel to one another and a number of curves that undergo pairwise narrowly avoided crossings. Adiabatic switching calculations show that the "parallel'' curves are associated with states within the classical resonance region, while the narrowly avoiding curves are associated with states that are outside this region. It is further shown that the curves describing resonance states are formed from zero-order nonresonant curves by the overlap of many avoided crossings. This reorganization of multiply intersecting lines into parallel curves reflects the classical reorganization of phase space at a resonance.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.466164
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  • 4
    Electronic Resource
    Electronic Resource
    Numerical study of semiclassical initial value methods for dynamics (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 4432-4445 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: We present numerical tests of five related semiclassical techniques for computing time-dependent wave functions. These methods are based on integral representations for the propagator and do not require searches for special trajectories satisfying double-ended boundary conditions. In many respects, the computational techniques involved resemble those of conventional quasiclassical treatments. Three of these methods result in globally uniform asymptotic approximations to the wave function. One such method, the treatment of Herman and Kluk, is found to be capable of especially high accuracy and rapid convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.466273
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  • 5
    Electronic Resource
    Electronic Resource
    Integral expressions for the semiclassical time-dependent propagator (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 100 (1994), S. 4377-4392 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.466320
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  • 6
    Electronic Resource
    Electronic Resource
    Semiclassical propagation for multidimensional systems by an initial value method (1994)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 101 (1994), S. 2250-2260 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A semiclassical initial value technique for wave function propagation described by Herman and Kluk [Chem. Phys. 91, 27 (1984)] is tested for systems with two degrees of freedom. It is found that chaotic trajectories cause a serious deterioration in the accuracy and convergence of the technique. A simple procedure is developed to alleviate these difficulties, allowing one to propagate wave functions of a moderately chaotic system for relatively long times with good accuracy. This method is also applied to a very strongly chaotic system, the x2y2 or "quadric oscillator'' model. The resulting energy spectra, obtained from the autocorrelation function of the wave function, are observed to be in good agreement with the corresponding quantal spectra. In addition, the density of states spectra, computed from the trace of the semiclassical propagator, are found to determine many individual energy levels of this system successfully.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.467665
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  • 7
    Electronic Resource
    Electronic Resource
    A quasi-classical approximation for autocorrelation functions (1988)
    s.l. : American Chemical Society
    The @journal of physical chemistry 〈Washington, DC〉 92 (1988), S. 602-607 
    Details
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1021/j100314a007
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  • 8
    Electronic Resource
    Electronic Resource
    Local ergodicity as a probe for chaos in quantum systems: Application to the Henon–Heiles system (1987)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 86 (1987), S. 4628-4647 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Classical chaos is usually accompanied by "local'' ergodicity—ergodic behavior in regions of phase space that are generally smaller than, but of the same dimensionality as, the energy surface. If there are strong bottlenecks in phase space impeding the relaxation to statistical equilibrium in the full region of ergodicity, it is often possible to define an approximate form of ergodic behavior in a smaller subregion where relaxation occurs temporarily but quickly. Such "pseudoergodicity'' is also a symptom of chaos. We use the presence of quantum behavior that mimics local ergodicity and pseudoergodicity as a probe for the influence of classical chaos on quantum dynamics. We show that the quantum analog of a pseudoergodic region in phase space is generally formed by superpositions of energy eigenstates. The superposition states spanning a given pseudoergodic zone have similar expectation values and small off-diagonal elements to other states with similar energy for a certain class of operators. We perform calculations which identify the ergodic regions for two versions of the quantum Henon–Heiles system. For the "more classical'' of these systems we find good agreement between the proportion of quantum states in pseudoergodic zones and the proportion of classical phase space occupied by chaotic trajectories. We also find that the quantum pseudoergodic regions can be identified with classical vague tori of the precessing and librating types. Different ergodic regions are separated from one another by what appear to be the quantum analogs of the precessor–librator separatrix and other classical bottlenecks. For the "less classical'' of the systems, we find that classical chaos is reflected in the quantum dynamics to a smaller, but still noticeable, degree.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.452704
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  • 9
    Electronic Resource
    Electronic Resource
    SEMICLASSICAL INITIAL VALUE TREATMENTS OF ATOMS AND MOLECULES (2005)
    Palo Alto, Calif. : Annual Reviews
    Annual Review of Physical Chemistry 56 (2005), S. 255-280 
    Details
    ISSN: 0066-426X
    Source: Annual Reviews Electronic Back Volume Collection 1932-2001ff
    Topics: Chemistry and Pharmacology , Physics
    Notes: This review describes some developments in the theory and application of the semiclassical initial representation for the treatment of the dynamical and static properties of atoms and molecules. The theoretical basis of initial value treatments for the propagator is discussed. A variety of useful alternative initial value expressions for the propagator and other quantities are presented as generalizations of the well-known Herman-Kluk approximation. Special emphasis is given to treatments that involve integration over only half the phase space variables. The recent development of semiclassical initial value expressions that are exact for specific, desired systems is reviewed and some of the implications are described.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1146/annurev.physchem.56.092503.141257
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  • 10
    Electronic Resource
    Electronic Resource
    Semiclassical tunneling in the initial value representation (1997)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 107 (1997), S. 2313-2328 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Tunneling in the one-dimensional Eckart system is treated by a semiclassical method that describes the S-matrix in terms of an integral over the initial momenta of real-valued classical trajectories. The results are found to be sensitive to a certain parameter γ which is expected to be essentially arbitrary for classically allowed processes. Analysis of the semiclassical error allows formulation of conditions for the validity of the tunneling treatment. This, in turn, leads to an explanation for the sensitivity of the results to γ and an understanding of how this parameter should be chosen. With an optimized choice, the semiclassical method is found to yield very accurate tunneling results even for probabilities as small as10−10. The relationship between the present method and the conventional uniform semiclassical treatment of barrier tunneling is discussed. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.475143
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