Tunneling in two-dimensional systems using a higher-order Herman–Kluk approximation

This paper deals with the critical issue of approximating the pre-exponential factor in semiclassical molecular dynamics. The pre-exponential factor is important because it accounts for the quantum contribution to the semiclassical propagator of the classical Feynman path fluctuations. Pre-exponential factor approximations are necessary when chaotic or complex systems are simulated. We introduced pre-exponential factor approximations based either on analytical considerations or numerical regularization. The approximations are tested for power spectrum calculations of more and more chaotic model systems and on several molecules, for which exact quantum mechanical values are available. The results show that the pre-exponential factor approximations introduced are accurate enough to be safely employed for semiclassical simulations of complex systems.

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“…The spectrum of each irreducible representation is reported in Fig. (12) with the same color code as above and for different approximations.…”

Section: Table (Ix)mentioning

confidence: 99%

The importance of the pre-exponential factor in semiclassical molecular dynamics

Liberto

Ceotto

2016

The Journal of Chemical Physics

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“…One needs approximations that keep track of quantum effects and scale well with system size. A number of methods that rely on time-evolution of a coherent state, such as QHD [13], Herman-Kluk (HK) propagator [14] and its higher order extensions [15,16], coupled coherent states (CCS) [17], matching-pursuit/split-operator Fourier transform (MP-SOFT) [18,19] and coherentstate path-integral (CSPI) [20], are effective for large systems. These methods have been successfully used only for slightly non-linear systems; in addition, these methods were derived to be applied to harmonic or slightly anharmonic system.…”

Section: Introductionmentioning

confidence: 99%

Signatures of discrete breathers in coherent state quantum dynamics

Igumenshchev

Ovchinnikov

Maniadis

et al. 2013

The Journal of Chemical Physics

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In classical mechanics, discrete breathers (DBs) - a spatial time-periodic localization of energy - are predicted in a large variety of nonlinear systems. Motivated by a conceptual bridging of the DB phenomena in classical and quantum mechanical representations, we study their signatures in the dynamics of a quantum equivalent of a classical mechanical point in phase space - a coherent state. In contrast to the classical point that exhibits either delocalized or localized motion, the coherent state shows signatures of both localized and delocalized behavior. The transition from normal to local modes have different characteristics in quantum and classical perspectives. Here, we get an insight into the connection between classical and quantum perspectives by analyzing the decomposition of the coherent state into system's eigenstates, and analyzing the spacial distribution of the wave-function density within these eigenstates. We find that the delocalized and localized eigenvalue components of the coherent state are separated by a mixed region, where both kinds of behavior can be observed. Further analysis leads to the following observations. Considered as a function of coupling, energy eigenstates go through avoided crossings between tunneling and non-tunneling modes. The dominance of tunneling modes in the high nonlinearity region is compromised by the appearance of new types of modes - high order tunneling modes - that are similar to the tunneling modes but have attributes of non-tunneling modes. Certain types of excitations preferentially excite higher order tunneling modes, allowing one to study their properties. Since auto-correlation functions decrease quickly in highly nonlinear systems, short-time dynamics are sufficient for modeling quantum DBs. This work provides a foundation for implementing modern semi-classical methods to model quantum DBs, bridging classical and quantum mechanical signatures of DBs, and understanding spectroscopic experiments that involve a coherent state.

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“…Many other methods have been developed for studying quantum effects in large nonlinear systems and calculating semiclassical correlation functions and spectra. A representative, but by no means complete, list includes Heller's frozen Gaussians [37], the Herman-Kluk propagator [38,39] and its higher order extensions [39,40], coupled coherent states [41], matching-pursuit/split-operator Fourier transform [42,43], and coherent-state path integrals [44]. These approaches are based on averaging over multiple trajectories or finding roots for a boundary value problem.…”

Section: Introductionmentioning

confidence: 99%

Quantized Hamilton dynamics describes quantum discrete breathers in a simple way

Igumenshchev

Prezhdo

2011

Phys. Rev. E

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We study the localization of energy in a nonlinear coupled system, exhibiting so-called breather modes, using quantized Hamilton dynamics (QHD). Already at the lowest order, which is only twice as complex as classical mechanics, this simple semiclassical method incorporates quantum-mechanical effects. The transition between the localized and delocalized regimes is instantaneous in classical mechanics, while it is gradual due to tunneling in both quantum mechanics and QHD. In contrast to classical mechanics, which predicts an abrupt appearance of breathers, quantum mechanics and QHD show an alternation of localized and delocalized behavior in the transient region. QHD includes zero-point energy that is reflected in a shifted energy asymptote for the localized states, providing another improvement on the classical perspective. By detailed analysis of the distribution and transfer of energy within classical mechanics, QHD, and quantum dynamics, we conclude that QHD is an efficient approach that accounts for moderate quantum effects and can be used to identify quantum breathers in large nonlinear systems.

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Tunneling in two-dimensional systems using a higher-order Herman–Kluk approximation

Cited by 11 publications

References 26 publications

The importance of the pre-exponential factor in semiclassical molecular dynamics

The importance of the pre-exponential factor in semiclassical molecular dynamics

Signatures of discrete breathers in coherent state quantum dynamics

Quantized Hamilton dynamics describes quantum discrete breathers in a simple way

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