Semiclassical quantization of a nonintegrable system: Pushing the Fourier method into the chaotic regime

The globally uniform semiclassical approximation for energy eigenstates developed by D. Zor and K. G. Kay ͓Phys. Rev. Lett. 76, 1990 ͑1996͔͒ is derived explicitly for the case of multidimensional systems and is applied to two test cases. The adiabatic switching approximation is used to obtain various quantities that enter the semiclassical expression. Two formulations of the method are examined: one requiring several trajectories for each desired state and another requiring only a single trajectory per state. The multitrajectory version yields accurate results for all states investigated ͑overlap between semiclassical and quantum eigenstates Ͼ0.98͒, including some influenced by classical chaos. The single-trajectory treatment, however, is more efficient, gives accurate results for regular states, and is even applicable for certain chaotic states, although the multiple-trajectory method is preferred in such cases. Despite the substantial resemblance of the present theory to the frozen Gaussian approximation ͑FGA͒, it is a true semiclassical approximation and is found to produce wave functions that are significantly more accurate than those obtained from the FGA for all states examined.

Section: Discussion and Summarymentioning

confidence: 99%

Globally uniform semiclassical wave functions for multidimensional systems

Madhusoodanan

Kay

1998

“…Heller used a cellular method 42 and tried to organize the dynamics of the system using classical trajectories in a hierarchical structure avoiding the root search. Also for a chaotic regime, an extension of the Fourier method has been proposed, 43 which uses the most dominant terms of the Fourier expansion to give a numerically truncated approximate torus, the semiclassical analog of the wave function.…”

Section: Discussionmentioning

confidence: 99%

A periodic orbit analysis of the vibrationally highly excited LiNC/LiCN: A comparison with quantum mechanics

Prosmiti

Farantos

Guantes

et al. 1996

Topology of the distribution of zeros of the Husimi function in the LiNC/LiCN molecular system J. Chem. Phys. 120, 6516 (2004); 10.1063/1.1665984 Local frequency analysis and the structure of classical phase space of the LiNC/LiCN molecular system By constructing continuation/bifurcation diagrams of families of periodic orbits of LiNC/LiCN system the spectroscopy and dynamics for this species are deduced and compared with accurate quantum mechanical calculations up to 13 000 cm Ϫ1 . The interesting phenomenon of the appearance of gaps in the continuation diagram of the principal family that corresponds to the bend motion is shown to occur in both isomers, LiNC and LiCN. Through semiclassical quantization a one to one correspondence of specific periodic orbits to certain eigenstates is demonstrated. One interesting example is the case of periodic orbits that are generated from a saddle-node bifurcation and describe rotations of the Li ϩ ion around the CN Ϫ fragment. The correspondence of these periodic orbits to regular rotating type eigenfunctions is shown, thus, demonstrating in a clear way the importance of the saddle-node bifurcations in locating localized wavefunctions in highly energized molecules.

“…This study is useful even without the application of a semiclassical method, because classical mechanics is often used to study the dynamics of molecules. The study of the present system is an extension of previous work, 3,19 because the degree of chaos is much greater and the amount of semiclassical information generated is complete over a large energy range. For example, in Ref.…”

Section: Introductionmentioning

confidence: 93%

“…13 Although many quantum mechanical methods for generating eigenvalues and eigenstates have been implemented recently, a good deal of early work explored semiclassical methods. [14][15][16][17][18][19] From a practical standpoint these methods were motivated by the idea that it would not be possible to apply quantum mechanical methods to highly excited states. However, although improvements in accuracy and ease of use have been made 18 and these methods have been extended into the chaotic region, 19 in recent years semiclassical methods have proven to be less applicable than quantum mechanical ones.…”

Section: Introductionmentioning

confidence: 99%

“…[14][15][16][17][18][19] From a practical standpoint these methods were motivated by the idea that it would not be possible to apply quantum mechanical methods to highly excited states. However, although improvements in accuracy and ease of use have been made 18 and these methods have been extended into the chaotic region, 19 in recent years semiclassical methods have proven to be less applicable than quantum mechanical ones. However, semiclassical methods together with quantum mechanical calculations, are useful for understanding quantum eigenstates and an important aid for assigning them, particularly highly excited eigenstates which may be very complicated.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Chaotic dynamics and approximate semiclassical quantization: Assigning highly excited vibrational eigenstates

Davis

1997

Accurate calculation and assignment of highly excited vibrational levels of floppy triatomic molecules in a basis of adiabatic vibrational eigenstates J. Chem. Phys. 95, 3456 (1991); 10.1063/1.461798Approximate constants of motion for classically chaotic vibrational dynamics: Vague tori, semiclassical quantization, and classical intramolecular energy flowThe classical dynamics of highly chaotic vibrational dynamics is studied via the nature of resonance zones. It is demonstrated that the flow into and out of the resonance zones is complicated with some trajectories spending less than one period outside before reentering, referred to here as ''virtual recrossing.'' Despite the strong chaos, an approximate semiclassical quantization method based on the resonance zone structure can be used to generate eigenvalues to within a few percent of the quantum values well into the chaotic regime. The semiclassical quantization allows for the assignment of highly excited, strongly mixed vibrational eigenstates.