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Molecular Physics

Dynamics of electron transfer in complex glassy environment modeled by the Cole–Davidson spectral density

Yun Zhou, Jiushu Shao, Haobin Wang

Published Apr 10, 2012• 11 citations
Publisher

Abstract

The dynamics of electron transfer reactions in a complex environment is investigated in the context of the spin-boson model with a bath characterized by the Cole–Davidson spectral density. Using the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method, the population dynamics of the two-level subsystem has been investigated in a broad physical regime. Upon changing various parameters, the simulation results exhibit either weakly damped coherent motion, incoherent decay, or localization. Transitions between these regimes are discussed in terms of several important physical parameters. Comparison of the exact ML-MCTDH simulations with the non-interacting blip approximation (NIBA) shows that the latter performs quite well in the nonadiabatic regime despite the complex multiple time scales the bath exhibits, but fails in the adiabatic and intermediate regimes where the relaxation of the bath is no longer significantly faster than the electron transfer process. In the nonadiabatic regime an interesting, semi-quantitative finding on electron transfer dynamics is discussed based on a simple relation between two parameters in the Cole–Davidson spectral density, the coupling strength and the fractional stretching exponent.

DOI
10.1080/00268976.2012.670281

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The dynamics of electron transfer reactions in a complex environment is investigated in the context of the spin-boson model with a bath characterized by the Cole–Davidson spectral density. Using the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method, the population dynamics of the two-level subsystem has been investigated in a broad physical regime. Upon changing various parameters, the simulation results exhibit either weakly damped coherent motion, incoherent decay, or localization. Transitions between these regimes are discussed in terms of several important physical parameters. Comparison of the exact ML-MCTDH simulations with the non-interacting blip approximation (NIBA) shows that the latter performs quite well in the nonadiabatic regime despite the complex multiple time scales the bath exhibits, but fails in the adiabatic and intermediate regimes where the relaxation of the bath is no longer significantly faster than the electron transfer process. In the nonadiabatic regime an interesting, semi-quantitative finding on electron transfer dynamics is discussed based on a simple relation between two parameters in the Cole–Davidson spectral density, the coupling strength and the fractional stretching exponent.
The dynamics of a two-level system coupled to a spin bath is investigated via the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Consistent with the previous work on linear response approximation [N. Makri, J. Phys. Chem. B 103, 2823 (1999)], it is demonstrated numerically that this spin-spin-bath model can be mapped onto the well-known spin-boson model if the system-bath coupling strength obeys an appropriate scaling behavior. This linear response mapping, however, may require many bath spin degrees of freedom to represent the practical continuum limit. To clarify the discrepancies resulted from different approximate treatments of this model, the population dynamics of the central two-level system has been investigated near the transition boundary between the coherent and incoherent motions via the ML-MCTDH method. It is found that increasing temperature favors quantum coherence in the nonadiabatic limit of this model, which corroborates the prediction in the previous work [J. Shao and P. Hanggi, Phys. Rev. Lett. 81, 5710 (1998)] based on the non-interacting blip approximation (NIBA). However, the coherent-incoherent boundary obtained by the exact ML-MCTDH simulation is slightly different from the approximate NIBA results. Quantum dynamics in other physical regimes are also discussed.
The dynamics of the spin-boson model at zero temperature is investigated using the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method.This method allows a numerically exact description of the dynamics of the spin-boson model in a broad range of coupling strengths.The results show the transition of the dynamics from weakly damped coherent motion to incoherent decay and finally to localization upon increase of the system-bath coupling strength.A detailed analysis of the incoherent decay for stronger coupling reveals that multiple timescales are involved in the dynamics.Furthermore, the applicability of the scaling limit for large characteristic frequencies of the bath as well as the validity of the non-interacting blip approximation are studied in some detail.
The transition from weakly damped coherent motion to localization in the context of the spin-boson model has been the subject of numerous studies with distinct behavior depending on the form of the phonon-bath spectral density $J\left(\ensuremath{\omega}\right)\ensuremath{\propto}{\ensuremath{\omega}}^{s}$. Sub-Ohmic (s<1s<1s<1) and Ohmic (s=1s=1s=1) spectral densities show a clear localization transition at zero temperature and zero bias, while for super-Ohmic (s>1s>1s>1) spectral densities this transition disappears. In this paper, we consider the influence of the phonon-bath spectral density on the nonequilibrium dynamics of a quantum dot with electron-phonon interactions described by the extended Holstein model. Using the reduced density matrix formalism combined with the multilayer multiconfiguration time-dependent Hartree approach, we investigate the dynamic response, the time scales for relaxation, as well as the existence of multiple long-lived solutions as the system-bath coupling changes from the sub- to the super-Ohmic cases. Bistability is shown to diminish for increasing powers of sss similar to the spin-boson case. However, the physical mechanism and the dependence on the model parameters such as the typical bath frequency ${\ensuremath{\omega}}_{c}$ and the polaron shift $\ensuremath{\lambda}$ are rather distinct.
The dynamics of a coupled electron–boson system is investigated by employing a multitude of the Davydov D 1 trial states, also known as the multi-D 1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz ). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D 1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron–nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.
The dynamical behavior of a two-level system coupled linearly to a dissipative bath of two-level systems whose tunneling splittings and coupling parameters are distributed according to an Ohmic spectrum is studied using numerical path integral methods as well as the noninteracting blip approximation for an equivalent harmonic environment characterized by a scaled spectral density. Comparisons with the behaviors of the conventional spin-boson model of similar coupling strength reveal a weaker quenching of coherence in the present case. Finally, the boundary in parameter space between coherent and incoherent dynamics is mapped.
A fundamental longstanding problem in studying spin models is the efficient and accurate numerical simulation of the long-time behavior of larger systems. The exponential growth of the Hilbert space and the entanglement accumulation at long times pose major challenges for current methods. To address these issues, we employ the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) framework to simulate the many-body spin dynamics of the Heisenberg model in various settings, including the Ising and XYZ limits with different interaction ranges and random couplings. Benchmarks with analytical and exact numerical approaches show that ML-MCTDH accurately captures the time evolution of one- and two-body observables in both one- and two-dimensional lattices. A comparison with the discrete truncated Wigner approximation (DTWA) highlights that ML-MCTDH is particularly well-suited for handling anisotropic models and provides more reliable results for two-point observables across all tested cases. The behavior of the corresponding entanglement dynamics is analyzed to reveal the complexity of the quantum states. Our findings indicate that the rate of entanglement growth strongly depends on the interaction range and the presence of disorder. This particular relationship is then used to examine the convergence behavior of ML-MCTDH. Our results indicate that the multilayer structure of ML-MCTDH is a promising numerical framework for handling the dynamics of generic many-body spin systems.