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University of Vienna

Institute for Theoretical Chemistry
and Structural Biology

Exciton (de)localization
by Wichard J. D. Beenken
and Pä:r Kjellberg

Contents
  1. Introduction
  2. Delocalization of excitons in dimers
  3. References

This work has been financially supported by Wenner-Gren Foundations

Introduction

Electronic excited states of interacting molecular systems has been an active research area since early work by Frenkel. The concept of collective delocalized excitations, excitons, was developed. Phenomena like Davydov splitting in molecular crystals, motional narrowing and superradiant decay in molecular aggregates have been explained by using the exciton theory. Particularly the excitons in photosynthetic light-harvesting antenna systems are in the focus of active research. Such issues as exciton relaxation, excitation delocalization versus localization, and the resulting spectroscopic signatures has been addressed. Already Frenkel recognized the important role of the nuclear motions in the concept of excitons. First of all, even a weak coupling between the electronic and nuclear degrees of freedom causes dephasing and population relaxation among exciton levels. For strong electron-phonon coupling, further phenomena like self-trapping of the exciton, also called polaron formation, occur. Various theoretical approaches addressing different aspects of exciton dynamics in antenna systems have recently appeared. For example Redfield relaxation theory, has been applied using model functions as well as experimental data for the spectral density to describe exciton relaxation and corresponding experimental observables in different antenna systems.

In order to describe simultaneously different stages of exciton dynamics we have recently developed a method based on nuclear dynamics for explicit vibrational modes combined with the surface hopping approach. In this method the complete dynamic process from dephasing and exciton relaxation to polaron formation (self-trapping), and eventually diffusion of the polaron can be described. This approach seems to be appropriate for application on excitonically coupled systems of arbitrary size, from the simple dimer to the extended photosynthetic antenna complexes.

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Delocalization of excitons in dimers1

There exists a large variety of dimeric molecular systems from interacting pair of guest molecules in a molecular mixed crystal to dimeric pigment complexes in biological systems e.g. the so called B820 antenna complex from purple bacteria or the special pair in the photosynthetic reaction center. Excited states and their dynamics in molecular dimers has been studied experimentally and theoretically by numerous authors. In the present work we calculate the potential surfaces more generally including also the heterodimer, where the transition energies of the two molecular sites are different.

Potential Surfaces

Schematic picture of coupled dimer

Schematic picture of the excitonic coupled dimer 

Using the model of two excitonically coupled electronic two-level systems with one-dimensional potential surfaces of the monomers given as displaced harmonic oscilators (see above) one obtains two excitonic states with adiabatic potential surfaces given by

     Equation of excitonic potentil surfaces

The resulting potential surfaces depend on two parameters: v which represents the strength of the excitonic coupling and Δ which represents the difference in the transition energies of the monomers, both in units of the Stokes shift of the monomers. For several combinations of these parameters the potential surfaces of the lower (U-) and the upper (U+) one exciton state has been calculated (see below). 

 

Point to parameter combinations v and Δ (assigned by · ) for a look on the lower exciton's potential surface U-.

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Nuclear motion

In order to describe the nuclear motion on the potential surfaces in a dissipative surrounding we use the classical Langevin equation 

Equation Langevin  

The fluctuating forces fi(t), and the damping rate g are connected by the dissipation fluctuation theorem as

Equation Forces

We have performed Monte-Carlo simulations of the nuclear motion to determine the time evolution of the statistical distribution of nuclear coordinates (see below).

Animation of nuclear motion

Exciton dynamics for the lower exciton in the case v = 0.5 and Δ = 0.1.

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Delocalization Length

The excitonic wave functions ψ+ and ψ-, respectively, given by the eigenvectors of the Hamiltonian containing exciton's potential energy U±, are delocalized over the dimer, i.e. they represents superpositions of the single-site excitations e1g2 and g1e2. We quantify this by the delocalization length L± using the participation ratio

,

Note that for the current potential energy U± defining the wave functions ψ±, the delocalization length L± depends on the nuclear coordinates as well as on the parameters v and Δ (see below). 

Point to the assigned parameter combinations (left) to see the dependences on the nuclear coordinates for L- (lower) and L+ (upper), respectively. (corresponding potential surfaces U- and U+  shown as overlaying white lines).

Averaged over a statistical distribution of nuclear coordinates one obtains effective delocalization lengths Leff± , which are shown below for the following cases:

  a b c d e h
v 1 1.5 1 0.5 0.5 0.5
D 0 0 1 0 1 0.25

Time evolution of Leff- (left) for lower exciton and  Leff + (right) for upper exciton 

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Surface hopping

Due to the non-adiabatic coupling, the upper exciton is not stable but will decay to the lower one* with the  rate 

 This results in a surface hopping from U+ to U-, which affects the nuclear motion and consequently the time evolution of the delocalization length (see below). 

*The back transfer from the lower to the upper exciton will be ignored for thermodynamic reasons.

Time evolution of Leff± including surface hopping from upper to lower exciton

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References

  1. W. Beenken, M. Dahlbom, P. Kjellberg, V. Sundström, and T. Pullerits: "Potential surfaces and delocalization of excitons in dimers." J. Chem. Phys. 117 (2002) 5810-5820.
     
  2. M. Dahlbom, W. Beenken, V. Sundström, and T. Pullerits: "Collective excitation dynamics and polaron formation in molecular aggregates." Chem. Phys. Lett. 364 (2002) 556-561.
     
  3. M. Dahlbom, W. Beenken, V. Sundström, and T. Pullerits: "Collective excitation dynamics in molecular aggregates: Exciton relaxation, self-trapping and polaron formation." In Biophysical Chemistry: Membranes and Proteins, eds. R. H. Templer and R. Leatherbarrow, Royal Society of Chemistry, London, (2001) 118-135.

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