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The Journal of Physical Chemistry B

Some New Classical and Semiclassical Models for Describing Tunneling Processes with Real-Valued Classical Trajectories

Jianhua Xing, Eduardo A. Coronado, William Hughes Miller

Published Apr 11, 2001• 15 citations
Publisher

Abstract

A model for describing barrier tunneling (or other classically forbidden processes) using purely real-valued classical trajectories is presented. The basic idea is to introduce an auxiliary degree of freedom that allows for fluctuations in the potential-energy surface for the original classical degrees of freedom. The model can be applied purely classically or better semiclassically (e . g ., via the initial value representation). Numerical results for 1D barrier tunneling are presented to illustrate the model.

DOI
10.1021/jp0046086

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A model for describing barrier tunneling (or other classically forbidden processes) using purely real-valued classical trajectories is presented. The basic idea is to introduce an auxiliary degree of freedom that allows for fluctuations in the potential-energy surface for the original classical degrees of freedom. The model can be applied purely classically or better semiclassically (e . g ., via the initial value representation). Numerical results for 1D barrier tunneling are presented to illustrate the model.
There has been a surprising degree of uncertainty about how, or even whether, the standard semiclassical theory of Van Vleck and Gutzwiller (VVG) can be used to treat the time-dependent dynamics of tunneling through barriers. At issue is how semiclassical theory describes the time evolution of tunneling. This uncertainty has hindered development of practical techniques for semiclassical calculations of multidimensional systems. It is shown here that the VVG theory can indeed be applied to barrier tunneling, but generally requires consideration of classical trajectories with both complex initial conditions and complex time paths, even for systems with one degree of freedom. The present approach is verified by applying it to the classically forbidden propagation of a wave packet through a one-dimensional barrier---a very basic problem that cannot be treated by the VVG theory without the modifications described here.
A semiclassical method of complex trajectories for the calculation of the tunneling exponent n systems with many degrees of freedom is further developed. It is supplemented with an easily implemented technique that enables one to single out the physically relevant trajectory from the whole set of complex classical trajectories. The method is applied to semiclassical transitions of a bound system through a potential barrier. We find that the properties of physically relevant complex trajectories are qualitatively different in the cases of potential tunneling at low energy and dynamical tunneling at energies exceeding the barrier height. Namely, in the case of high energies, the physically relevant complex trajectories describe tunneling via creation of a state close to the top of the barrier. The method is checked against exact solutions of the Schrödinger equation in a quantum-mechanical system of two degrees of freedom.
A semiclassical method of complex trajectories for the calculation of the tunneling exponent in systems with many degrees of freedom is further developed. It is supplemented with an easily implementable technique, which enables one to single out the physically relevant trajectory from the whole set of complex classical trajectories. The method is applied to semiclassical transitions of a bound system through a potential barrier. We find that the properties of physically relevant complex trajectories are qualitatively different in the cases of potential tunneling at low energy and dynamical tunneling at energies exceeding the barrier height. Namely, in the case of high energies, the physically relevant complex trajectories describe tunneling via creation of a state close to the top of the barrier. The method is checked against exact solutions of the Schrodinger equation in a quantum mechanical system of two degrees of freedom.
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 as 10−10. The relationship between the present method and the conventional uniform semiclassical treatment of barrier tunneling is discussed.
The time independent semiclassical treatment of barrier tunneling has been understood for a very long time. Several semiclassical approaches to time dependent tunneling through barriers have also been presented. These typically involve trajectories for which the position variable is a complex function of time. In this paper, a method is presented that uses only real valued trajectories, thus avoiding the complications that can arise when complex trajectories are employed. This is accomplished by expressing the time dependent wave packet as an integration over momentum. The action function in the exponent in this expression is expanded to second order in the momentum. The expansion is around the momentum, p0(*) , at which the derivative of the real part of the action is zero. The resulting Gaussian integral is then taken. The stationary phase approximation requires that the derivative of the full action is zero at the expansion point, and this leads to a complex initial momentum and complex tunneling trajectories. The "pseudo-stationary phase" approximation employed in this work results in real values for the initial momentum and real valued trajectories. The transmission probabilities obtained are found to be in good agreement with exact quantum results.
A central challenge to the semiclassical description of quantum mechanics is the quantum phenomenon of "deep" tunneling. Here we show that real time classical trajectories suffice to account correctly even for deep quantum tunneling, using a recently formulated semiclassical initial value representation series of the quantum propagator and a prefactor free semiclassical propagator. Deep quantum tunneling is effected through what we term as coherent classical paths which are composed of one or more classical trajectories that lead from reactant to product but are discontinuous along the way. The end and initial phase space points of consecutive classical trajectories contributing to the coherent path are close to each other in the sense that the distance between them is weighted by a coherent state overlap matrix element. Results are presented for thermal and energy dependent tunneling through a symmetric Eckart barrier.