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The European Physical Journal Plus

Dissipative electronic nonadiabatic dynamics within the framework of the Schrödinger–Langevin equation

Ching-Hwa Ho, Chia‐Chun Chou

Published Sep 1, 2021• 5 citations
Publisher
DOI
10.1140/epjp/s13360-021-01963-2

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Resonant tunneling of electrons through a dissipative barrier is studied within a generalized spin–boson model. The model accounts for a nonlinear inelastic coupling between the electron and harmonic (bath) nuclear modes inside the barrier. In particular, we study the case in which the nuclear frequencies are similar to the decay rate of the electronic resonance states, as may appear in electron transport through narrow molecular barriers. Two cases are considered. In the case of a single responsive mode numerically exact wave-packet dynamics shows that the inelastic coupling increases the time delay of the tunneling electron in the barrier and the decay process out of the barrier is characterized by nonexponential steps. In the case of a multimode (Ohmic) bath an approximate treatment of the dynamics, based on temperature dependent nonlinear Langevin–Schrödinger equations, implies that the trapped electron wavepacket is fully solvated in the barrier due to irreversible dissipation at low temperatures. An interpretation of the solvation process in terms of the mean-field approximation is given and discussed.
Ignoring thermal fluctuations of the environment, taking into account only its dissipative effects, free propagation of a Gaussian wavepacket is studied in the framework of the linear Caldirola-Kanai (CK) equation and non-linear equations, the Schrodinger-Langevin (SL) equation, known as Kostin equation, and the Schuch-Chung-Hartmann (SCH) equation. By a Gaussian ansatz for the probability density one obtains two equations, one for the evolution of the center of the wave packet which is just the classical Langeving equation and one for the evolution of the width of the wavepacket. This last equation has different forms in different approaches having only an analytic solution in the CK and SCH frameworks. Computations show that for a given friction, the width of the wavepacket increases with time in all approaches. In a given time, it reduces with friction in both CK and SL approaches revealing localization effects of dissipation, while has opposite behavior in the SCH approach. Furthermore, energy expectation value and its time-derivative is computed and compared in all approaches. It is shown that the rate of energy is given by the expectation value of momentum field in the SL framework. Finally, transmission of a Gaussian wavepacket from a rectangular barrier is numerically studied in the context of CK and compared to the non-dissipative case. The transmission decreases considerably with the dissipation.
Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.