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TitleSimulation of energy transfer dynamics in light-harvesting complexes
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Table of Contents
List of Abbreviations
	Photosynthesis and harvesting of light
	Fenna-Mathews-Olson complex
	Phycoerythrin 545 antenna
	Spectroscopy and EET in LH complexes
Theory of open quantum systems
	Excitonic Hamiltonian
		Two-state model
	Reduced density matrix dynamics
		Harmonic bath
	Ensemble-average wave-packet dynamics
		Molecular Dynamics Simulations
	Electronic excitation energies Em
		Semi-empirical methods
		Quantum mechanics/molecular mechanics (QM/MM) approach
	Excitonic pigment-pigment couplings Vmn
		Point-dipole approximation (PDA)
		TrESP method
		Poisson-TrESP method
		Spectral Density
	Outline of results and discussions
Juxtaposing density matrix and wave-packet dynamics methods
	Model Hamiltonian for exciton transfer
	Excitation transfer dynamics
		Reduced density matrix approach
		Ensemble-averaged wave-packet dynamics
	Comparing density matrix and wave-packet dynamics: two-site system
	Temperature-corrected wave packet approach
The FMO complex in a glycerol-water mixture
	Comparison to known approximations for the spectral densities
Vibrations in PE 545 and FMO antenna systems
	Spectral density
	Excitation dynamics
Population dynamics in the PE545 complex
Appendix SI: Vibrations in PE 545 and FMO antenna systems
Appendix Excitation Transfer and 2D Spectra of FMO
List of Tables
List of Figures
List of Publications
Document Text Contents
Page 1

Simulation of energy transfer
dynamics in light-harvesting



Mortaza Aghtar

A thesis submitted in partial fulfillment
of the requirements for the degree of

Doctor of Philosophy

in Physics

Approved Dissertation Committee:

Prof. Dr. Ulrich Kleinekathöfer
(Jacobs University Bremen)

Prof. Dr. Thomas Heine
(Jacobs University Bremen)

Dr. Carles Curutchet
(University of Barcelona)

Date of Defense: June 4th, 2015

Page 84

78 Chapter 5. Vibrations in PE 545 and FMO antenna systems

5.3 Spectral density

The next step in the process of obtaining spectral densities is to calculate the autocorrelation

function which, in principle, is different for each pigment. The autocorrelation function is

determined using the energy gaps ∆Ej(ti) at time steps ti. The energy gap autocorrelation

function for pigment j is given by[94]

Cj(ti) =

N − i


∆Ej(ti + tk)∆Ej(tk) . (5.1)

In the present case, the autocorrelation functions are calculated over time lengths of 2 ps. This

length was seen to generate all modes in the spectral densities and, as can be seen in Fig. S2,

the autocorrelation functions have decayed to zero within 1 ps. To generate correlation

functions over 2 ps, trajectory pieces of 4 ps length are necessary. The autocorrelation

functions of the 180 trajectory pieces (720 ps divided by 4 ps) are averaged to obtain the final

result for each pigment. The autocorrelation functions are shown in Fig. S2. Furthermore,

shown in Fig. S3 are the autocorrelation functions for the PE545 and the FMO complexes

averaged over all pigments of the respective systems. On average the autocorrelation function

in case of the FMO trimer shows smaller fluctuations than in the other cases.

The spectral density Jj(ω) of pigment j can be expressed as [63, 145]

Jj(ω) =



dt Cj(t) cos(ωt) (5.2)

where β = 1/(kBT ). In some previous studies [61, 89, 94] we used the prefactor 2 tanh(βh̄ω/2)/πh̄

instead of its high-temperature limit βω/π. This previous variant is slightly inconsistent

with replacing the real part of the quantum autocorrelation function with its classical high-

temperature counterpart and the present version has been shown by Valleau et al. [63] to

yield more consistent results. This change in the prefactor affects only the high-frequency

regime of the spectral density.

The spectral densities of the individual pigments were determined using Eq. 6.1. To

increase the quality of spectral densities, i.e., to reduce the noise level, we included an ex-

ponential cutoff function for the autocorrelation functions starting at 1 ps to damp them

slowly to zero at 2 ps. The resulting spectral densities for the 8 pigments of the PE545

complex are shown in Fig. 6.3. This figure focuses on the low-frequency range which is of

utmost importance for the EET since the differences between excitonic states are usually in

Page 85


5.3. Spectral density 79









PEB 50/61C

2 2.2 2.4 2.6

0 0.02 0.04 0.06 0.08 0.1

w [eV]




BChl 1

1.4 1.5 1.6

0 200 400 600 800

w [cm




Figure 5.3: Spectral densities of the energy gap autocorrelation function for three bilins
(top panel) and for a BChl in FMO (bottom panel). The solid lines indicate the results
with the standard QM/MM coupling while the dashed lines the results with the environment
frozen. The insets display the respective distributions of excited state energy gaps along the

the corresponding energy range.

Overall the spectral densities of the pigments in PE545 and FMO exhibit much similarity.

The coupling strengths to the environment follows similar trends and values. Due to different

molecular structures of the chromophores, differences in the internal modes arise as expected.

However, surprising is the effect of freezing the point charges of the environment to one

snapshot. As reported already earlier [62], the low frequency part of the spectral density for

the FMO complex results from the fluctuations in the environment. This can be nicely seen

in Fig. 6.3 that compares the results obtained with a standard QM/MM coupling and with a

frozen environment. In the latter case the spectral density is close to zero and only for larger

frequencies some small contributions from internal modes become visible.

The low frequency behavior of the energy gap autocorrelation function is drastically

different for the PE545 complex. For some pigments there is almost no change when freezing

the movement of the point charges in the environment during the QM/MM step while for

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