Energy relaxation and transfer in excitonic trimer
Introduction
The very symmetric arrangement [1] of BChl molecules with short distances between them brought up the usage of the exciton concept in the description of the energy transfer in the LH2 ring-shaped subunit of the antenna complex of Rps. acidophila. The energy transfer in this purple bacteria is likely to involve initially a coherent stage, in which the excitation is delocalized over several molecules. Localization takes place due to the static and dynamic disorder [2]. The excited-state dynamics are measured using, e.g., one- and two-color pump-spectroscopy, transient grating and three-pulse photon-echo spectroscopy. Studies have been conducted at both low temperatures and room temperature.
While the influence of the static disorder can be theoretically taken into account by repeating the calculation for another set of the stochastic local energies, one is generally forced to deal with a very complicated Liouville equation for the whole system-bath density matrix σ to describe the exciton interacting with phonon bath. That is why various types of convolutional or convolutionless dynamical equations for the exciton density matrix ρ (bath degrees of freedom are traced off) have been often used to describe coherence effects in the exciton transfer.
In the past, two different approaches have been developed:
Working on the basis of exciton eigenstates of the molecular aggregate, the Redfield equations in the secular approximation (henceforth referred to as the Redfield equations) have been often used [2].
On the other hand, the static and dynamic disorder destroys correlations between the phases of distant molecular sites and typical electronic properties are therefore local, and an intuitive physical picture should then be based on the properties of wave packets. A local real-space description (the dynamical equations for the exciton density matrix in the local site basis) could be more appropriate [3].
The main message of our work is that it could be risky only to transform the Redfield equations from the eigenstate basis to the local-site basis.
In a previous paper [5], we investigated exciton transfer and relaxation in a symmetric dimer. We have shown how the proper dynamical equations (first developed by Čápek) for the exciton density matrix ρ(t) written in the local state basis resemble, but substantially enrich, the equations used in the Haken–Strobl–Reineker Stochastic Liouville equations method (HSR-SLE) [4]. In this paper we continue our analysis (from Ref. [6]) by concentrating on exciton transfer and relaxation in a symmetric trimer with an energy spectrum (a two-fold degenerate upper level and a non-degenerate lower level) similar to the three-level model used for an explanation of the optical properties of the LH2 subunit [7].
Section snippets
Hamiltonian
We shall be dealing with just one exciton in the symmetric cyclic trimer interacting with the phonon bath. The Hamiltonian then consists of three partswhere H0 is the exciton part, Hb the bath part and Hint describes the local and linear exciton phonon coupling.
The first part (where , an are creation and annihilation operators of exciton on site n and J is the transfer integral) iswhich gives for J<0 eigenenergies EI=−2|J|, EII=EIII=|J|
Dynamical equations for the exciton density matrix
To follow exciton transfer and relaxation one has to obtain the time dependence of the exciton density matrix ρ.
Results
To compare these different models of exciton transfer in the local-site representation and in the eigenstate representation, we have to transform the equations of Čápek and HSR-SLE into the eigenstate basis and the Redfield equations back to the local-site representation.
From such a comparison, we conclude:
(i) There are nonzero elements for i≠j in the Redfield model contrary to the Čápek and HSR-SLE models. This means (in the HSR-SLE notation) . But the parameters should be
Acknowledgements
This work has been partially funded by the project GAČR 202-03-0817 of the Czech Grant Agency.
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