Semiclassical electron transfer: Zusman equations versus Langevin approach

doi:10.1016/S0301-0104(01)00292-0

Chemical Physics

Volume 268, Issues 1–3, 15 June 2001, Pages 151-164

https://doi.org/10.1016/S0301-0104(01)00292-0 Get rights and content

Abstract

The model of a donor–acceptor electron transfer (ET) being coupled to an overdamped reaction coordinate is studied semiclassically. Starting from an archetypal ET Hamiltonian, two different routes of the semiclassical approximation are critically analyzed and mutually compared. A first path proceeds along the Caldeira–Leggett form for the master equation for the reduced density matrix of “electron+reaction coordinate” which is cast in its Wigner phase–space representation. Integrating over the momentum of the reaction coordinate then yields (in the overdamped limit) the so-termed Zusman ET-equations. The alternative route starts from the formally exact quantum Heisenberg–Langevin equations. The corresponding derivation of the equations of motion for the observables involves a sort of a mean-field semiclassical approximation. The final result are nonlinear stochastic differential equations (SDE) of the Langevin type. We compare the results of these two approaches, both in the absence and in the presence of external time-dependent driving fields. Our findings are that both methods yield good agreement for the description of symmetric ET. In contrast, however, the SDE fails to describe the ET dynamics when a strong static bias is present. The inclusion of a strong time-periodic field ET-manipulation improves the Langevin approximation scheme, providing reasonable good agreement between both routes.

Introduction

A wealth of basic research work to investigate open quantum systems has been put forward in recent years and applied to numerous applications in the fields of chemistry and physics [1], [2], [3], [4]. A prominent example with ever increasing activity addresses the problem of charge transfer in a dissipative environment. There exist numerous models in the literature designed to describe electron (or proton) transfer [2]. A particularly appealing approach is based on the familiar spin-boson model [1] which provides a pragmatic, yet still realistic formulation for the physics of a dissipative donor–acceptor electron transfer (ET) [2].

Often, the dissipative influence of the environment allows for a semiclassical description, while at the same time the full quantum nature of the electron dynamics must be accounted for. Such a type of the mixed quantum-classical description is presently very much on vogue [5], [6], [7], [8], [9], [10], [11]. The literature offers several such methods, but at the same time providing no guide of how to single out a most useful and advantageous approximation scheme. In this work we compare two popular such approaches. The Zusman approach, introduced in Section 3, and the nonlinear Langevin model, presented in Section 4, are not solely restricted to describe ET in condensed phases; these provide a generic scheme to model a few level system coupled to a dissipative oscillator. As such these two schemes carry the potential for even broader applications in modeling the dynamics of open quantum systems.

Section snippets

Electron transfer model

Let us consider the following archetypal Hamiltonian $H ̂_{ET} (t)= 12 [V_{1} (x ̂,t)−V_{2} (x ̂,t)] σ ̂_{z} + 12 Δ σ ̂_{x} + 12 p ̂^{2} m +V_{1} (x ̂,t)+V_{2} (x ̂,t) 1 ̂ + H ̂_{B}$ which is widely used to describe the ET processes in condensed media [2], [12]. In Eq. (1), $σ ̂_{z} :=|1〉〈1|−|2〉〈2|$ and $σ ̂_{x} :=|1〉〈2|+|2〉〈1|$ are the pseudo-spin operators expressed via the localized donor, |1〉, and acceptor, |2〉, states; $1 ̂$ is the 2×2 unity matrix. Furthermore, the diabatic electronic states $V_{1,2} (x ̂)$ constitute the Born–Oppenheimer potentials for the nuclear

Zusman equations

The first approach to the above stated problem utilizes the reduced density matrix method. In a first step, the master equation for the reduced density matrix $ρ ̂$ of the “electron+reaction coordinate” system is derived in the coordinate representation. This task can be accomplished using either the real-time path integral approach [12], or by the cumulant expansion method [19]. The resulting master equation reads within the semiclassical approximation of the bath [12], [23]: $d d t ρ ̂ (t)=− i ℏ [H ̂_{ET} (t), ρ$

Langevin approach

Because the Zusman equations present an analogue of the Smoluchowski-like equations we pose the question: does there exists an equivalent formulation in terms of SDEs (Langevin approach)? To answer this challenge we start from the Heisenberg equations of motion for the Hamiltonian (1) and integrate over the thermal bath variables (see, e.g., Refs. [1], [3], [4], [17]). The final result reads $d d t σ ̂_{x} (t) =− 1 ℏ (ϵ(t)+V_{1} (x ̂)−V_{2} (x ̂)) σ ̂_{y} (t),$ $d d t σ ̂_{y} (t) =− 1 ℏ Δ σ ̂_{z} (t)+ 1 ℏ (ϵ(t)+V_{1} (x ̂)$ $−V_{2} (x ̂)) σ ̂_{x} (t),$ $d d t σ ̂_{z} (t) = 1 ℏ Δ σ ̂_{y}$

Numerical results and discussion

Let us test these preliminary conclusions numerically. The ET process within the considered model can be characterized by the following phenomenological parameters: (i) the electronic coupling Δ; (ii) the static energy bias ϵ₀; (iii) (medium's) reorganization energy E_r; (iv) (medium's) autocorrelation time τ, and (v) the temperature T. We have performed our analysis with the following set of parameters being typical for a nonadiabatic ET regime: $Δ=10 cm^{−1}$ , $E_{r} =500 cm^{−1}$ , $τ=1 ps$ , and $T=300 K$ . The

Conclusion

With this work we primarily have performed a detailed comparison between two familiar, although very different approaches in order to implement the semiclassical approximation for the sluggish reaction coordinate dynamics of a charge transfer dynamics. These two schemes relate to a generalization of the Zusman approach and the nonlinear semiclassical Langevin dynamics. In doing so, we have considered both an undriven ET dynamics and a time-periodic ET-manipulation via external, generally strong

Acknowledgements

This work has been supported by the Deutsche Forschungsgemeinschaft within the Schwerpunktsprogramm Zeitabhängige Phänomene und Methoden in Quantensystemen der Physik und Chemie, HA1517/14-3, and within the Sonderforschungsbereich 486, project A10. We also would like to thank M. Morillo, C. Denk, and J. Casado-Pascual for helpful discussions and for providing us with the preprint of Ref. [8].

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Semiclassical electron transfer: Zusman equations versus Langevin approach

Abstract

Introduction

Section snippets

Electron transfer model

Zusman equations

Langevin approach

Numerical results and discussion

Conclusion

Acknowledgements

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Chem. Phys. Lett.

Chem. Phys.

Physica A

Chem. Phys.

Chem. Phys. Lett.

Chem. Phys.

Rev. Mod. Phys.

J. Chem. Phys.

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J. Chem. Phys.

J. Chem. Phys.

J. Chem. Phys.

J. Chem. Phys.

J. Chem. Phys.

J. Chem. Phys.

J. Chem. Phys.

J. Chem. Phys.