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Physical Review

Quantum Rate Theory for Solids. II. One-Dimensional Tunneling Effects

Jerome H. Weiner, Yehuda Partom

Published Nov 15, 1969• 69 citations
Publisher

Abstract

The effects of tunneling are included in a one-dimensional rate theory which utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets (coherent states). An approximate tunneling formula for such wave packets is developed for a piecewise quadratic potential representing a well followed by a barrier; it is based on an exact solution for a Gaussian wave packet moving on a parabolic barrier. A new method for the numerical solution of the time-dependent Schr\"odinger equation is presented and used to check this approximate tunneling formula. The addition of tunneling to the effect of quantum statistics gives a rate formula which approximates the classical result at high-temperature levels and approaches a constant rate determined by tunneling at very low-temperature levels.

DOI
10.1103/physrev.187.1134

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The effects of tunneling are included in a one-dimensional rate theory which utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets (coherent states). An approximate tunneling formula for such wave packets is developed for a piecewise quadratic potential representing a well followed by a barrier; it is based on an exact solution for a Gaussian wave packet moving on a parabolic barrier. A new method for the numerical solution of the time-dependent Schr"odinger equation is presented and used to check this approximate tunneling formula. The addition of tunneling to the effect of quantum statistics gives a rate formula which approximates the classical result at high-temperature levels and approaches a constant rate determined by tunneling at very low-temperature levels.
A theory for rate processes in solids is presented which includes the effects of quantum statistics, tunneling, and the NNN-dimensional character of the problem. It utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets to describe the conditions of thermal equilibrium in an NNN-dimensional potential well. An exact solution is found for the motion of Gaussian wave packets on a second-degree NNN-dimensional potential surface. This solution is valid also when the initial principal directions of the wave packet differ from those of the potential surface so that the problem is not separable. It is used to compute the tunneling probability for a Gaussian wave packet on a second-degree NNN-dimensional potential surface containing a saddle point. This probability depends only on the wave-packet characteristics in the saddle-point direction, and this facilitates its incorporation in the rate expression. Since the initial characteristics of the wave packets of the ensemble are determined by the potential-well character, and their subsequent tunneling probability is determined by the saddle-point character, the theory includes the effect of a difference in orientation between the two sets of principal directions. For an artificial two-dimensional problem formulated specifically to study this aspect of the process, an orientation difference is found to produce large effects only at low temperatures where tunneling is important.
A model for interstitial diffusion is studied in which the impurity atom is regarded as a Brownian particle constrained to move in a single direction and subject to a periodic potential. A quantum mechanical theory is developed for the rate at which atoms leave a given potential well. The theory makes use of an ensemble of minimum-uncertainty wave packets to describe thermal equilibrium in the well and computes the tunneling probability through the potential barrier on the basis of the Schr\"odinger-Langevin equation in order to incorporate the effect of an interacting heat bath. The result of the interactions is to reduce the departure rate at a given temperature level, but the qualitative nature of the rate expression remains unchanged and predicts Arrhenius plots with straight-line behavior at high-temperature levels and curvature due to tunneling at low-temperature levels. The dynamics of the wave packet after it leaves a well is studied by computer solution of the Schr\"odinger-Langevin equation and it is found that the persistence of motion decreases with increasing strength of interaction with the heat bath.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.
Quantum tunneling effects of Gaussian wave packet in one- and two-dimensional double well potentials are investigated using quantum Liouville equation for time evolution of Wigner distribution in phase space. It is shown that a trajectory-based solution of this problem can be constructed by the entangled trajectory ensemble simulating the evolving quantum state. Quantum effects arise in this approach as a breakdown of the statistical independence of the trajectory ensemble. © 2014 Wiley Periodicals, Inc.
A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.
The detailed calculation of the dynamics of a chemical system is usually not considered due to the size and cost of the computation. It is thus useful to examine various approximation methods. Such methods first need to be tried out on simple systems, like one-dimensional motion. Here two approaches to approximating the solutions of the Schrödinger and von Neumann equations by single time-dependent Gaussians are explored and contrasted, explicitly for one-dimensional barrier penetration. The first approach, in which no tunneling occurs, is local in nature and characterized by an expansion of the equations of motion to second order about the average position of the Gaussian wave packet or about the average position and momentum of the Gaussian Wigner function. This approach was first introduced by Heller [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. Here both Heller’s approach and a more general truncation method are considered. Indeed tunneling can be incorporated if second-order terms in the quantal von Neumann equation are included. However, the resulting dynamics is unstable for kinetic energies where the exact wavepacket (and Wigner function) splits into nonnegligible parts that represent reflection and transmission. In contrast, the second approach is a global method which is obtained by applying appropriate closure approximations to the equations of motion for the first- and second-order position and momentum expectation values. This method allows tunneling and is stable at all kinetic energies. It is also possible to approximate the global equations in such a way that the local equations are obtained.