Theoretical basis of energy and electron transfer

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Theoretical basis of energy and electron transfer
EPA summerschool, Egmond, June 2003
Gert van der Zwan (zwan_at_few.vu.nl) Department
of Analytical Chemistry and Applied
Spectroscopy Vrije Universiteit, Amsterdam
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Contents.
Day 1 Polarization properties of solvents

• Statics and dynamics of dielectrics
• Onsager, Lorentz, Debye, Lippert-Mataga
• Dynamic Stokes shift

Day 2 Reaction rates and Marcus theory

• From transition state theory to solvent
  coordinates
• Non-equilibrium free energy surfaces
• Electron transfer and Marcus theory

Day 3 Molecules in solution and energy transfer

• How molecules feel the world and each other
• Excitonic interaction, coherent transfer
• Dipole interaction and the Förster rate

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Excitonic interaction and excitation transfer (1)
Bacterial Photosynthesis (1)
Source M. Brederode, TU Delft
Electron and proton transfer processes in the
bacterial reaction center.
Source Holten, Washington University in St. Louis
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Excitonic interaction and excitation transfer (2)
Bacterial Photosynthesis (2)
A recent proposal (Jungas et. al. 1999) for LH1
organization is shown here.
Source Theoretical Biology group, University of
Illenois at Urbana-Champaign
This allows the UQ to leave the RC. Likely the
PufX protein plays a role here.
NB atomic resolution structural models for LH1
are based on LH2 structure
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Excitonic interaction and excitation transfer (3)
Bacterial Photosynthesis (3)
The excitation energy is transfered between the
LH2 rings, to the LH1 antenna, and finally to the
reaction center. Below are estimates of the times
involved.
Mostly the Förster transfer mechanism is invoked,
although sometimes it is argued that coherent
energy transfer is the mechanism.
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Excitonic interaction and excitation transfer (4)
Bacterial Photosynthesis (4)
Top view of LH2 of Rps. acidophila, note the
9-fold symmetry
B800
ß
a
B850
Side view of LH2 from Rsp. molischianum, note the
8-fold symmetry.
A nice collection of images can be found on
Richard J. Cogdells webpages www.gla.ac.uk/ibls/
BMB/rjc/lh2galery.htm
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Excitonic interaction and excitation transfer (5)
Bacterial Photosynthesis (5)
Source PROMISE database
From this point on the protons take over
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Excitonic interaction and excitation transfer (6)
Excitation and energy transfer (1)
hole transfer
h?
DA-
D-A
excitation transfer 1
excitation transfer 2
Donor (D)
Acceptor (A)
DA
DA
Excitation
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Excitonic interaction and excitation transfer (7)
Excitation and energy transfer (2)
Molecules can transfer excitation energy in a
number of ways (emission-absorption, Dexter and
Förster mechanism), but an other possibility is
to describe them as excitonically coupled.

• Dexter overlap of wavefunctions
• Förster overlap of spectra, dipole interaction
• Excitonic dipole interaction, exciton splitting
• Role of disorder (diagonal, correlated)

single excited states
h?
Excitonic interaction
excitonically coupled dimer
monomer 1
monomer 2
Exercise find differences and similarities
between excitation and electron transfer
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Excitonic interaction and excitation transfer (8)
The Förster mechanism (1)
Model for Förster transfer between donor and
acceptor molecules. kA and kD are decay
constants, kT is the Förster transfer rate.
Relaxation within the vibrational manifolds is
assumed to be rapid.
If we use units such that hc1, everything
can be expressed in cm
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Excitonic interaction and excitation transfer (9)
The Förster mechanism (2)
The donor decays mainly from the ground
vibrational level in the electronically excited
state. The transition moments are Sm?D, where Sm
is the Franck-Condon overlap, and ?D is
the electronic transition moment.
The acceptor is mainly excited from the
vibrational and electronic ground state. The
transition moments are Sn?A, where Sn is
the Franck-Condon overlap, and ?A the electronic
transition moment.
Vmn
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Excitonic interaction and excitation transfer (10)
The Förster mechanism (3)
Proportional to the emission spectrum of the donor
Proportional to the absorption spectrum of the
acceptor
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Excitonic interaction and excitation transfer (11)
The Förster mechanism (4)
Spontaneous emission
The spontaneous emission rate in free space is
given by
This was derived by Glauber and Lewenstein,
Phys. Rev. A, 43, (1991), 467. Note that it
contains the Onsager field factor!
Exercise Why is ?n Sn2 1?
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Excitonic interaction and excitation transfer (12)
The Förster mechanism (5)
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Excitonic interaction and excitation transfer (13)
Hopping or relaxation?
Excitation Energy Transfer (EET)
Förster mechanism
(this is way off)
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Excitonic interaction and excitation transfer (14)
The Förster mechanism (6)
Very weak coupling at 10 nm two 6.3 D
dipoles have V0.2 cm-1
Angular dependence random orientation ?
2/3 head-to-tail ? -2
parallel ? 1
Distance dependence strong 1/R6
Spectral overlap weak between same
molecules (Stokes) suitable labels can be
found
Sytnik et.al., PNAS, 93, (1996), 12959
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Excitonic interaction and excitation transfer (15)
Single-molecule excitation transfer measurements
on Kinesin
excitation transfer
donor
acceptor
Kinesin Two-headed motor protein. Motility is
driven by confor- mational changes induced by
ATP hydrolysis. The conformational dynamics are
being studied using Förster Excitation
Transfer.
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Emission of single FRET pair 1. Excitation
Transfer 2. Acceptor bleached 3. Donor bleached
1
2
donor image
acceptor image
L.C. Kapitein, E.J.G. Peterman, C.F. Schmidt
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Excitonic interaction and excitation transfer (16)
Excitonic interaction (1)
Excitonically coupled dimer. The monomers 1 and 2
have similar excitation energies, and the
interaction strength V12 is much larger than for
Förster transfer
In B820 (a BChla dimer) the interaction strength
is about 300 cm-1, and the orientation of the
dipoles is almost parallel
Qy
Qx
BChla absorption
B820 absorption (2 BChla)
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Excitonic interaction and excitation transfer (17)
Excitonic interaction (2)
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Excitonic interaction and excitation transfer (18)
Excitonic interaction (3)
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Excitonic interaction and excitation transfer (19)
Excitonic interaction (4)
Some examples
head-to-tail orientation
sandwich orientation
?1
?-2
(Dark)
(Superradiant)
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Excitonic interaction and excitation transfer (20)
Excitonic interaction (5)
4V12
Transition dipole moments in the B850 ring
Excitonic states of the B850 Qy region
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Excitonic interaction and excitation transfer (21)
Excitonic interaction (6) spectra
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Excitonic interaction and excitation transfer (22)
Excitonic interaction (7) spectra
Absorption and CD spectra of LH2 from Rps.
acidophila. LH2 is also called B800-850 due to
the main absorption features in that region. The
800 band is due to the ring of BChla located at
the periphery of the antenna complex, but it
overlaps with a weak band (the high-exciton band)
of the B850 BChlas. This overlap is
sometimes seen as a cause of the 800-850
transfer, although the carotenoids are also
implicated.
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Excitonic interaction and excitation transfer (23)
Redfield Theory (1)
single exciton manifold
Dynamics relaxation to lowest excitonic state is
often described by Redfield theory
Assumptions coupling to external fields
is weak relaxation of the system
is slow compared to relaxation processes
in the medium
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Excitonic interaction and excitation transfer (24)
Redfield Theory in five easy steps
Formal solution in the interaction picture
1
(in the interaction picture things only change
because of the interaction)
Expand to second order
2
Average over external fluctuations
3
(average of the fluctuating fields is zero)
Differentiate and separate operators from fields
4
(correlation functions do not depend on the
initial time)
Change integration limits, and time dependence or
?I
5
(correlation functions decay rapidly on the time
scale of changes in ? )
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Excitonic interaction and excitation transfer (25)
Redfield Theory (3)
Redfield equation
? relaxation matrix
Example (NMR) spin ½ relaxation in fluctuating
magnetic fields
?0
Similar expressions hold for x, and y
magnetization
Exercise find differences and/or similarities
between this derivation and the one for Förster
transfer.
For those interested in a more detailed
derivation a .pdf file can be obtained from me
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Excitonic interaction and excitation transfer (26)
Redfield Theory (4)

• System feels the solvent, but not the other way
  around
• Interaction between solvent (bath) and system is
  weak
• Time scales are different bath relaxes much
  faster than system
• Relaxation of the system depends on bath
  correlation functions probed at the
• systems frequency
• Used mainly in NMR relaxation, but more recently
  also in exciton relaxation.

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Excitonic interaction and excitation transfer (27)
Hopping or relaxation?
?i,i1
?i,i-1
Isolated pigments
Förster transfer Weak coupling Hopping from
pigment to pigment Dynamics determined by the
master equation
i2
i-1
i
i1
DISORDER?
localizes excitations on pigments? or makes
excitonic states more local?
Excitonically coupled pigments
Exciton relaxation Sronger coupling Weak coupling
to environment Dynamics governed by the
Redfield equation.
Exercise think of an experiment that could show
the difference
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Excitonic interaction and excitation transfer (28)
The fundamental equation
Light fields linear and non-linear optics Expand
Liouville equation to desired order, ?(1),
?(3) Absorption, fluorescence, CD pump
probe,transient gratings, photon echoes, ......
Random fields Slowly varying static
disorder correlated disorder. Rapidly varying
relaxation Independent of the state of the system
Static external fields Stark spectroscopy
Reaction fields Self consistent interaction
with bath. Statics several QM
methods Dynamics ? Brownian oscillator (cl)
Exercise determine if the Liouville equation is
more fundamental than the Schrödinger equation.
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Is this all usefull? (well, maybe not, but it is
fun)
Excited state double well potential. Shown is the
potential surface, with the five lowest wave
functions cal- culated using the Fourier Grid
Hamiltonian method. Upon excitation with a short
pulse, a time dependent proton density is created.
Probability density for finding the proton at
position x if the system is excited with a 15 fs
light pulse, obtained by solving the Liouville
equation with the 15 lowest proton states on
the left.
Literature, references, and other information
with the exercises/tutorial