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The Journal of Chemical Physics

Extraction of tunneling splittings from a real time semiclassical propagation

Vladimir A. Mandelshtam, M. Ovchinnikov

Published Jun 8, 1998• 38 citations
Publisher

Abstract

The long-standing question of whether the tunneling effects can be recovered from a real time semiclassical propagation is addressed. Even at very low energies below the barrier for symmetric one- and two-dimensional double wells the tunneling splittings are accurately computed using harmonic inversion of the time cross-correlation matrix by the filter-diagonalization method. The time propagation is carried out using the Herman–Kluk propagator.

DOI
10.1063/1.476416

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The Journal of Chemical Physics•Dec 1, 1978•43 citations
The long-standing question of whether the tunneling effects can be recovered from a real time semiclassical propagation is addressed. Even at very low energies below the barrier for symmetric one- and two-dimensional double wells the tunneling splittings are accurately computed using harmonic inversion of the time cross-correlation matrix by the filter-diagonalization method. The time propagation is carried out using the Herman–Kluk propagator.
We study to what extent propagation of wave packets through potential barriers can be described semiclassically in real time. By analyzing the semiclassical approximation to the exact path-integral representation of the quantum-mechanical time evolution operator, we show that in the long-time sector it is dominated by quasiclassical fluctuations associated with caustics. Contrary to the description of tunneling in imaginary times, simple tunneling paths even for complex energies do not exist. Our findings lead to an extended semiclassical propagation scheme for wave-packet dynamics which is numerical efficient and accurately describes deep incoherent tunneling through static and externally driven barrier potentials.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.
Two methods have been widely employed for computing tunneling rates in a double-well potential, one based on a transmission function, the second on energy splitting. Semiclassical calculations (Brickmann and Zimmermann) show that the transmission method leads to lower rates than the splitting method. It is shown here without employing the semiclassical approximation that a more accurate relation between the energy splitting and the transmittion function may be obtained by using a decomposition of the stationary states into scattering states. This relation is then used to provide an analytical basis for the usual heuristic picture in which the particle oscillates with classical frequency in one well and has a probability of tunneling to the other well equal to the transmission function in each classical period of oscillation. It is concluded that the transmission method should give more meaningful results, particularly in situations where interactions of the system with a heat bath have significant effects in the tunneling times predicted by the splitting method. This is in accord with previous work in which both methods were used in an attempt to fit experimental data for hydrogen diffusion in niobium.