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J. Phys. Chem. B 2000, 104, 9276-9287
Theory of Excitation Energy Transfer in the Intermediate Coupling Case. II. Criterion for
Intermediate Coupling Excitation Energy Transfer Mechanism and Application to the
Photosynthetic Antenna System
Akihiro Kimura, Toshiaki Kakitani,* and Takahisa Yamato
Department of Physics, Graduate School of Science, Nagoya UniVersity, Chikusa-ku, Nagoya 464-8602, Japan
ReceiVed: February 14, 2000; In Final Form: June 22, 2000
We developed a theory of excitation energy transfer (EET) which is applicable to all the values of the coupling
strength U in the presence of homogeneous and inhomogeneous broadening. In constructing the theory, we
adopted a decoupling procedure corresponding to the factorization by a two-time correlation function of the
excitation transfer interaction in the integro-differential equation of a renormalized propagator. We also assumed
that the two-time correlation function decreases exponentially with time. Under these assumptions, we could
handle our theory nonperturbatively and analytically. We derived formulas of criteria among exciton,
intermediate coupling, and Förster mechanisms. We exploited a novel method for determining the EET rate
applicable to all the mechanisms from Förster to exciton. Then, we obtained compact formulas for the EET
rate and the degree of coherency involved in the EET. We demonstrated how the exciton state is destabilized
by the presence of inhomogeneity in the excitation energy of the constituents. The theory was applied to a
light-harvesting system LH2 of photosynthetic bacteria.
1. Introduction
Excitation energy transfer (EET) is a physically, chemically,
and biologically important phenomenon. Above all, EET in the
initial process of photosynthesis is quite significant for the light
1
called the Förster mechanism.9 Förster formulated the EET rate
based on Fermi’s Golden rule, expressing the coupling by the
transition dipole-transition dipole interaction which is applicable for the donor and acceptor that are capable of optically
allowed transitions.9 Dexter10 extended the Förster mechanism
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classic hopping mechanism
(Förster theory)
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Coherent EET in many natural antennae from different organisms:
LHC II (superior plants), RC (purple bacteria), FMO (green bacteria), PBP (cryptophytes)
E.Collini et al., Nature,
463, 644, 2010
H.Lee et al.,Science,
316, 1462, 2007
…
T.R.Calhoun et al., JPCB,
113, 16291, 2009
G.S.Engel et al., Nature,
446, 782, 2007
…
quantum biology
Crescente interesse verso effetti
quantistici nel light harvesting ed in
altri processi di rilevanza biologica
chains, exteng a rigid back-
a good solvent (chloroform) where the polymer
adopts an extended chain configuration, and (ii)
aqueous suspensions of polymer nanoparticles
(NPs) formed by individual collapsed chains (23).
All experiments were performed in continuously
flowing dilute solutions at 293 K (11). The spectra
of both samples (fig. S3) exhibited the typical
features known for this class of conjugated polymer (24, 25).
The conjugated polymer NPs showed little
decay of anisotropy during t (Fig. 2A), whereas
the well-solvated MEH-PPV chains in chloroform
Slices through the experimental surfaces at
t = 0 are plotted in Fig. 3A. It is evident that the
anisotropy decays during T more for NPs than for
the MEH-PPV in chloroform solution because interchain interactions mediate efficient EET (20).
Despite that observation, we saw no coherence
anisotropy decay for NPs, which suggests that
fluctuations on unconnected conformational subunits are uncorrelated (see below), as is normally
assumed. In Fig. 3B, experimental data points as
a function of t are plotted for T = 0 and are
compared to simulations (solid lines). Owing to
Downloaded from www.sciencemag.org on Janu
cause it is often
pproximation).
s recorded as a
ys (t and T ) in
ransient grating
opulation time,
cs such as EET
ay t, introduced
s a time period
|0〉〈d| between
tes (11). During
dephasing (free
T from |0〉〈d| to
completely disdecay by monthe time delay
ents, which we
EETalso occurs
.
ue to study cougated polymer
-1,4-phenylenepolymers have
ause they comanding mechanth the versatile
f functional orin and among
l concern as a
enching, and is
how excitation
n of chain conConjugated poms for seeking
ause the ultraes is governed
nd localization
orientation of
sing anisotropy
,-O!#!"'&!"!#$%&'#(")*!#+&,O!21)'#%&
E.Collini et al., Science, 323, 369, 2009
Poly(phenylene vinylene) chain
open chain
conformation
tightly coiled
conformation
conformational
subunit
inter chain transfer
conjugation break
intrachain transfer
Fig. 1. Example of single-chain conformation of a poly(phenylene vinylene) conjugated polymer,
referred as the defect cylinder conformation. Conformational disorder produces a chain of linked
chromophores (or conformational subunits) outlined conceptually by boxes. The intrachain EET (migration
along the backbone) is the predominant mechanism when the polymer chain assumes an open, extended
conformation, typical for solutions in good solvents such as chloroform. On the other hand, interchain
interactions (hopping between segments in close proximity) are dominant for tightly coiled
configurations, polymer nanoparticles, or films.
16 JANUARY 2009
VOL 323
SCIENCE
Figure 4. (a) Packing diagram for c-P6 3 T6. Front (b) and side (c)
views of individual molecules of c-P6 3 T6 in the crystal with mean plane
fitted through six zinc-centers (red dashed line). Hydrogen atoms, aryl
groups, and solvent molecules are omitted for clarity.
www.sciencemag.org
Hayes et al., Science, 340, 1431, 2014
corresponding angle between meso carbon atoms C(10) and
C(20) (mean γ = 177.9! ( 0.5!; Figure 5c). A similar curved
distortion is evident in the porphyrin tetramer nanobarrel
reported recently by Osuka and co-workers.21 Most of the strain
in c-P6 3 T6 is distributed among the acetylenes. The averages of
the C(sp2)!CαtCβ and CαtCβ!Cβ bond angles for the six
nonequivalent CtC units in c-P6 3 T6 deviate significantly from
values found for linear butadiynes in the CSD (Figure 5d).
However the distortion in these acetylenes is less than that found
Figure 5. Comp
those of structu
Database: (a) Zn
out-of-plane dista
atom porphyrin
centroid of four
(sp2)!CαtCβ (
(gray bars) and cy
indicate the value
inequivalent buta
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