Beyond Förster resonance energy transfer in
biological and nanoscale systems
J. Phys. Chem. B, 2009, 113 (19), pp 6583–6599
DOI: 10.1021/jp900708f
David Beljonne, Carles Curutchet, Greg Scholes, Bob Silbey
David@averell.umh.ac.be
Laboratory for Chemistry of Novel Materials
University of Mons-Hainaut, Belgium
Warwick, August 09
Taking Advantage of Excitation Diffusion in
Conjugated Polymers Based...
... displays
... solar cells
15 x 10 cm, 300 dpi, 1 MB
Photo: Philips
Color purity
h+
e-
Charge generation
Al
ITO
Al
ITO
University of Linz
... (bio)chemical sensors
CH3
NO2
NO2
NO2
TNT
LUMO
LUMO
HOMO
HOMO
Polymer
Swager and co., MIT
Polymer + TNT
Förster Resonant Energy Transfer (FRET)
Resonant Energy Transfer (RET)
D*
A*
kda
hn
D
Virtual Photon exchange
A
D=donor, high-energy chromophore
A=acceptor, low-energy chromophore
Förster Resonant Energy Transfer (FRET)
Fermi Golden Rule
2
k FRET 

Vdd = point dipole-point
dipole interaction
2Vdd
Qi
Qf
sVdd
2
J
J
Spectral Overlap
Förster Resonant Energy Transfer (FRET)
1 R0 
2
2
 0   
sVdd J
 D R 

6
kFRET

2 0
9000
(ln
10
)

D
6
4
R0 
I
(

)

(

)

d
D
A
5
4

128  N A n
0
Energy transfer efficiency:
T 
1
R 6
1 ( )
R0
FRET = ‘Spectroscopic Ruler’
Beyond Förster Resonant Energy Transfer
RET
• Electronic coupling poorly described by point
dipole approximation
• Screening by the environment: s  1/n2
• Shared vibrational modes
GRET
• Energy transfer between weakly interacting
aggregates
CRET
• Intermediate coupling & coherences
Point Dipole and Transition Densities
Vmn
Transition density approach
m* 
Vmn 


Z
a
i

a
 ia  m m mim  m
D
m
ia
n
i ,a
j ,b
m
n
Z
Z
 ia j b 2 bm jm | anin  - bnim | am jn   P  2 J  K  P
Rmn  lm, ln
Rmn > lm, ln
TDM
PDM
Rmn
µm
Vmn
1

4 0
 m (i )  n ( j )

rij
i
j
m
n
Vmn 
 
1
 m3 n
4 0
Rmn
µn
  ˆ m  ˆ n  3( ˆ m  Rˆ mn )(ˆ n  Rˆ mn )
Transition density approach: Wiesenhofer et al., Adv. Funct. Mat. 15, 155 (2005) ;
Beljonne et al., J. Chem. Phys. 112, 4749 (2000); see also Scholes, Rossky,
Barford… Line dipole model: Sundström, Pullerits and co
Examples of transitions densities calculated using quantum-chemical methods. A transition density,
defined as eg  e g , represents the superposition state of ground (g) and excited (e) states. Note
how they follow the shape of the chromophore in each case. (a) A bis-perylene derivative. (b)
Tryptophan. (c) A bilin from cryptophyte photosynthetic antenna systems. (d) The S2 state of a
carotenoid from the LH2 complex of a purple bacterium.
Breakdown of the Point Dipole Approximation
S1
Sn
Z
S0
Y
OPVn
C60
X
S1-S0 (~14D; x-polarized)
S0-S9 (~0.1D)
S0-S15 (~0.6D)
z-polarized
S0-S25 (~0.6D)
xy-polarized)
Breakdown of the Point Dipole Approximation
kRET (1012s-1)
Dyad
Calculateda
Experimentb
OPV2–MPC60
6.6 (1.1)
2.9
OPV3–MPC60
2.3 (0.3)
2.1
OPV4–MPC60
0.5 (0.1)
1.1
Calculated and measured energy transfer rates (kRET) in OPVn-MPC60 dyads. The
numbers in between parentheses correspond to the Förster rates and the point dipole
approximation.
PDA in Nanoscale Systems
Squared electronic couplings predicted by the point-dipole model (circles) and by a multicentric
transition density approach (triangles) as a function of the donor-acceptor center-to-center
separtation. (a) Two 3.9 nm diameter wurtzite CdSe quantum dots aligned parallel with respect to
their crystallographic c-axis. (b) Two parallel (7,5) semiconducting single wall carbon nanotubes.
Through-Space versus Through-Bond Coupling
ACCEPTOR
DONOR
IEFPCM reaction
potential, with
implicit and explicit
solvent effects
Solvent Screening Effects
s  A exp( R)  s0
The molecule’s transition density polarizes
the surrounding solvent through the
interaction with the electric field indicated.
This three-dimensional polarization can be
represented as effective solvent charges
collected on a surface enclosing the
molecules, plotted on the right-hand-side.
Solvent screening factors, s, predicted for
various chromophore pairs of lightharvesting antennae proteins. The Förster
value, 1/n2, is indicated by the lower
horizontal line, and the Onsager value,
3/(2n2+1), is the upper line.
Spectral Overlap and the Structure of the ‘Bath’
0
1
0

f D  t  =exp    g D
 D

f A  t  =exp    g A
 A
2
1
Quantum-chemical calculations
of ground-state and excitedstate geometries + normal mode
decomposition:
2
emission
absorption
Energy
Jmn Displaced (an)harmonic oscillator model

g 
Q2
2
Electron-phonon coupling
    g2
Reorganization energy

Ql


 1  exp   g D

 D


 1  exp   g A

 A
   2n    
 
    
 
2
D
2
2n  A
2
2

n 
D


  exp  i t     n     1 exp   i t   
D
D


 1 exp  i  A t  

   exp  i t     n     
n 
A

A
A
D

Spectral Overlap and the Structure of the ‘Bath’
kDA




|VDA |2
00
D
A 2

Re  dt expi DA t exp g  g  2n(  )  1

 




D
A 2
exp g  g  n(  )exp(i  t)  n(  )  1exp(i  t)
 

Separate baths for donor and acceptor (Förster)
For each l, either glD or glA = 0:

k DA
2
~ 00
~
2

VDA Re  d FA ( DA
  )FD ( )



2
2
VDA J

Spectral Overlap
Spectral Overlap and the Structure of the ‘Bath’
Shared vibrational modes: A Marcus-like picture
J NF  Re

 dt exp(i
0 0
DA
t) f A (t) f D (t) f DA (t)

1

2


 d  d F̃ (
1

2
00
DA
0 0
 1 ) F̃D ( DA
  2 ) F̃DA ()

(a) Same sign displacements
l < lD + lA
A
(b) Opposite sign displacements
l > lD + lA
Continuous distribution of low-frequency vibrational modes
(Gaussian lineshape)
Förster

 0 0
 
D 2
A 2



(
g
)

(
g
)

 DA  




2




exp 
D 2
A 2
D 2
A 2
 2 ( g  )  ( g  )  2n ( )  1 
( g  )  ( g  )  2n ( )  1
 




J F GB 
 




Non-Förster

 0 0
 
D
A 2



(
g

g
)

 DA  




2




exp 
D
A 2
D
A 2
 2 ( g   g  )  2n ( )  1 
( g   g  )  2n ( )  1
 




J NF GB 
 




Continuous distribution of low-frequency vibrational modes
(Gaussian lineshape)
Förster
Non-Förster: sgn (gD)= sgn (gA)
Non-Förster: sgn (gD)= - sgn (gA)
14000
A
D
g =1.0; |g |=2.0
12000
Overlap
-1
max(cm )
10000
A
D
g =1.0; |g |=1.6
8000
6000
4000
A
D
g =1.0; |g |=1.2
2000
0
-2000
0
2000
4000
6000
8000
10000
DA(cm-1)
Envelope of the effective spectral overlap as a
function of the D-A electronic for different values
and signs of the electron-phonon couplings. The
tick line is the Förster result. The thin and dashed
lines correspond to displacements with opposite
and same signs, respectively. T=300 K and
wl=400 cm-1.
0
1
2
3
D
g
4
5
6
D-A electronic gap maximizing the spectral overlap
factor as a function of the excitation-phonon
coupling on the donor in the case of: (i) Independent
modes (Förster case, thick line); (ii) one single nonlocal vibrational mode (non-Förster case, with
opposite-sign displacements, thin line); (iii) same as
(ii) for same-sign displacements (dashed line).
wl=400 cm-1 and T=300K.
Energy Diffusion in the Weak Coupling Regime
Weak Coupling Regime
Fermi Golden Rule
d
Pm  t  = dt
 k
nm
P  t  - knm Pn  t  + Pm m1 
mn m
Pauli Master Equations (PME)
Monte Carlo simulations
k mn
2 2

Vmn J mn

Spectral Overlap
Homogeneous and Inhomogeneous Disorder
In mutichromophoric donor-acceptor systems, we must calculate
spectral overlaps using homogeneous line broadening and then average
the electronic coupling–spectral overlap function over static disorder!
(a)
Experimental
absorption
spectrum of the LH2 lightharvesting complex isolated from
the purple bacterium Rps. acidophila
(room temperature). The B800 and
B850 bands are labeled. (b)
Calculated absorption spectra of the
B850 ring only for individual LH2
complex (77 K). Notice that each
complex is different. (c) Views of
the LH2 structure. The 18
bacteriochlorophyll-a molecules that
comprise the B850 ring are colored
pink, while the 9 in the B800 ring
are blue. The carotenoids (rhodopin
glucoside) are colored gray.
GFT + proper averaging
over disorder
Calculated distributions of rates, i.e., calculated for each LH2 complex in an
ensemble. The red curve is the FRET distribution. The blue curve shows the
calculations of Jang, et al. using a version of GFT
Modeling transport with Gaussian disorder
d
dt
atomistic
Förster and PME
Pm  t  = -   kmn Pm  t  - knm Pn  t  + Pm  t  m1 
nm
k mn 
Coarse graining
z
2 2
Vmn J mn

Miller-Abraham and MC

 E  E 
6
1  rF   exp  
,
  E , E , r      
kT 

r 
1,
r
u ( E , E , r )  6 ln
 rF
x
E  E
E  E
  (E  E)
 
kT

Equilibrium transport
D
3D lattice
continuum
1
1
   r 2 
exp    u    r 2 
6
6


2
  g ( E ) exp( E / kT ) exp(  u  ( E ))dE 
3
 



lD  D  rF 
g
(
E
)
exp(

E
/
kT
)








 E2 
Nt
g E 
exp   2 
s 2
 2s 
1/ 2
Exciton diffusion length
Evguenia Emelianova, Stavros Athanasopoulos, Phys. Rev. B, in press
Molecular Aggregates as Donors and Acceptors
  1   2  / 2

 
   ( 1   2 ) / 2  0

 

   ( 1   2 ) / 2  2 


; Ag Vdd Dg ; Ae  1;Ag Vdd Dg ; Ae   2 ; Ag Vdd Dg ; Ae / 2
V=0
V0!
(a) Illustration of an aggregate donor–acceptor system. Strong coupling within each of the
donor and acceptor groups forms molecular exciton states that are the effective donor and
acceptor states in GRET. (b) Depiction of the model dimer system discussed in the text. (c)
Transition densities calculated for the upper P+ and lower P– exciton states of the special pair
dimer in the photosynthetic reaction center of the purple bacterium Rhodobacter sphaeroides.
Weak coupling: Vmn~10-40 cm-1
Excitonic couplings and transport regime in
molecular crystals: The acenes
J-aggregate
‘Hopping between layers’
Strong coupling: Vmn~100-300cm-1
Evguenia Emelianova
Energy transport in oligoacenes
E
Layer 1
Layer 2
Boltzmann
RET
multiple pathways (k=0  0 for N  )
13
10
Partial cancellation
at short d
12
Rate of energy transfer, s
-1
10
11
10
10
10
9
10
8
10
Distance between layers, A T=300 K
11,46
15
20
30
50
100
7
10
6
10
1
10
100
N, number of molecules in the layer
1000
Dipole-dipole
at large d
Energy transport in oligoacenes
6
13
10
4
3
12
10
2
10
100
11
10
Number of molecules
-1
1
RET, s
Exponent
5
1/d3-4
10
10
9
10
1/d6
8
10
7
10
Number of molecules
in the layers
1
4
12
40
80
176
260
416
800
PBC
6
10
0,01
0,1
inverse distance between layers, A
-1
Energy transport: Role of dimensionality
k ET
1 d 0n

 dn
d0= critical distance (Förster radius for n=6)
d
1/d6
1/d4
1/d3
Hill et al., Phys. Rev. B 69, 041303 (2004); H. Kuhn, J. Chem. Phys. 53, 101 (1970);
Shaw et al., Phys. Rev. B 78, 245201 (2008); Scully et al., Adv. Mater. 19, 2961 (2007)
Energy transport in oligoacenes
2
Random walk: LD  D  k RET  r  
1000
950
900
Delocalization inplane increases Ld
out-of-plane by a
factor ~2-3
850
anthracene
Diffusion length, A
800
750
700
650
600
pentacene
550
500
450
T=300 K
Distance between layers, A
11,46
15
20
400
350
300
Requires coherent
domains of 10-100
molecules
250
200
1
10
100
N, number of molecules
k ET
1d

 d
6
0
6
anthracene
d0=R0= 4,2nm
k ET
1 d 03

3
 d
d0= 90nm
Energy Diffusion in the Strong Coupling Regime
Vmn
Enn
Exciton states:
Polaron transformation:
Pauli Master equations:
Pereverzev, Bittner, JCP 125, 104906, 2007
Delocalization in Conjugated Polymers
Polyfluorenes
thin films
Laura Herz, PRL 98, 027402, 2007
MEH-PPV, diluted solutions
Greg Scholes, Toronto
Modeling MEH-PPV Single Chains
Random growth algorithm
chromophores
Exciton matrix
Gil Claudio and Eric Bittner
University of Houston
S1,S1
 E1,S
V
0
1, 2
1

Site
 VS21,,S11 E 2,S
1 energy


Electronic
coupling
E1,Sn

Sn,Sn

V
2,1


VS11,Sn
,2





VSn,Sn
1, 2 
E 2,Sn 
Polarization Anisotropy Decay
Eigenstate representation
Site representation
Smaller radius
of gyration
Short time scale (~10ps): fast, short-range energy transfer
Long time scale (~200ps): slow, long-distance hopping
Bittner et al., in preparation
Intermediate Coupling and Coherent Dynamics
h /V >> tdecoh
h /V << tdecoh
Weak coupling
Strong coupling
The time-dependent population density of the electronically excited
acceptor state, |a>, evolves in time after initial preparation of |d>
according to:
a (t)  Tr  d V (t ) a aV ( t ) d d 
i t


V (t )  expT   H (t )dt 
  0

Intermediate Coupling and Coherent Dynamics
Calculations of the evolution of acceptor population density as a function of time after
photo-excitation of the donor. U is the electronic coupling between donor and acceptor.
The line width parameter is related to the decoherence time via g = /tc. A linear
relationship between acceptor population and time indicates the validity of the Fermi
golden rule expression for RET.
Excitation Delocalization in OPV Chiral Stacks
Dodecane solutions at T<Tm
Left helices for 4R enantiomer
Rotation angle <f>~-14° (Dreiding)
Hoeben et al., Angew. Chem. 43, 1976 (2004)
Holstein Model
normal mode frequency
normal mode displacement
(l2~S)
H   0  bn† bn   0   (bn†  bn ) | n  n |
n vibration

m
n
vibronic
2
(

D

J
)
|
m

n
|




0
 mn n mn
00
n
electronic coupling
electronic
molecular basis
diagonal disorder
Absorption, PL, CD and CPL Spectra
Weak coupling limit
angle = -14°
s=0.12 eV, l0~4.5
Absorption, PL, CD and CPL Spectra
Abn, CD  angle = -14°
PL, CPL  s=0.12 eV and l0=4.5
Coherent Exciton Size
Nuclear distortion field
Ncoh ~ 2-3 units
Exciton coherence function
Electronic Excitations in OPV Chiral Stacks:
A Simplified Picture
Ninc~50
Disorder : l0=4.5
EV coupling: Ncoh~2
t<1ps
Vertical excitation,
no disorder
Thermalized excitation,
disorder
Spano et al. JACS 129, 7044 (2007)
Jean-Luc Brédas, Jérôme Cornil, Evguenia Emelianova, Johannes
Gierschner, Emmanuelle Hennebicq, Jasper Knoester, Benedetta
Mennucci, Frank Spano, Bernard Van Averbeke, Alison Walker
Overdamped Brownian + Discrete Oscillators
Interchain Delocalization in Conjugated Polymers
MEH-PPV, diluted solution
Tieneke Dykstra & Greg Scholes, Toronto
Probing Interchain Delocalization in MEH-PPV
Inter-chromophore delocalization
TDM rotation
PR angle (°)
40
1.6
30
1.4
20
1.2
1.0
10
0.8
0
22000 24000
24000 26000
26000 28000
28000 30000
30000 32000
32000 34000
34000 36000
36000
22000
-1
Energy
(cm -1))
Energy (cm
Probing Interchain Delocalization in MEH-PPV
TDM rotation angle of ~17° (anisotropy~0.35) due to self-trapping:
Localization alone not enough to explain anisotropy data
Modeling Absorption, PL and Dynamics
Absorption and emission lineshapes
Spectral diffusion
Exciton states:
Master equations:
Overlap probability
Spectral density
Didraga et al. JPCB, 110, 18818 (2006)
Absorption and PL
Polarization Anisotropy Decay
R=154 A
R=154 A
Excitonic Couplings and Transport Regime in CP
Amorphous
Cristalline
Hopping
Wave-like
Hopping !!!
J k  k ,1 V k , 2
where

1
e ik p  p ,1

N P

1
ik q
k ,1 
e
  q ,1
N q
  
1
1
J k   e ik ( p q )  p ,1 V  q , 2 
N p ,q
N
k ,1 

 eikl )  p,1 V  p l ,2
p ,q
 
 ˆ  ˆ
Positive
interactions
 and
  3( negative
R )(  R )
J  e

cancel out in Rthe infinite chain limit

ikl
k
l
1
2
1 l
3
l
2
l


1 2
12
2

l e  2 2 3/ 2  3l 2 2 5/ 2 
l  d  
 l  d 
that can be rewritten after continuum approximation

ikl



2
1
1
l2
ikl cos 


J k  1 2
dl d 
3
2
2 5/ 2 
  l 2  d 2 3/ 2
2 A 0 0
l  d  

 0 for k  0
Excitation Delocalization in OPV Chiral Stacks
Dodecane solutions at T<Tm
Left helices for 4R enantiomer
Rotation angle <f>~-14° (Dreiding)
Hoeben et al., Angew. Chem. 43, 1976 (2004)
Building up the Hamiltonian
normal mode frequency
normal mode displacement (g2~S)
H    bn† bn   g   (bn†  bn ) | n  n |
n vibration

m
n vibronic
2
(

D

V
)
|
m

n
|



g
 mn n mn
00
 
n
electronic coupling
electronic
molecular basis
diagonal disorder
! Identification of ‘chromophores’ in CP !
Absorption, PL, CD and CPL Spectra
Weak coupling limit
angle = -14°
s=0.12 eV, l0~4.5
Absorption, PL, CD and CPL Spectra
Abn, CD  angle = -14°
PL, CPL  s=0.12 eV and l0=4.5
Coherent Exciton Size
Nuclear distortion field
Ncoh ~ 2-3 units
Exciton coherence function
Electronic Excitations in OPV Chiral Stacks:
A Simplified Picture
Ninc~50
Disorder : l0=4.5
EV coupling: Ncoh~2
t<1ps
Vertical excitation,
no disorder
Thermalized excitation,
disorder
Spano et al. JACS 129, 7044 (2007)
Generating Morphologies
Single pentacene crystals
Discotic liquid crystals
Force Field Molecular Dynamics
Polymer chains (isolated, solution, bulk)
Generating Morphologies
Polymer Random Growth Algorithm
O
O
O
O
Quantum-chemical potential
energy (bending, torsion, etc)
Keep track of the atomic
details
Through-Space versus Through-Bond Coupling
f
GS
400
TB+TS
E(cm-1)
300
ES
200
100
acceptor-acceptor
donor-donor
donor-acceptor
TB sensitive
to confinement
effects
mostly TS
0
0
10
20
30
40
(°)
50
60
70
80
90
Through-Space versus Through-Bond Coupling
8
400
acceptor-acceptor
donor-donor
200
CT character (%)
TB+TS
4-state
model
-1
E(cm )
300
1st ES, acceptor-acceptor
1st ES, donor-donor
7
100
mostly TS
6
TB+TS
5
4
3
mostly TS
2
1
0
0
0
10
20
 E1*

V12
H 
t
t

30
40
50
(°)
60
t 

*
E2 t
t 

1 2
t ECT
0 
2 1 
t 0 ECT

70
80
90
V12 t
t  t0 cos( )
0
10
20
30
40
50
(°)
60
70
80
Strong (weak) mixing
between local and CT
configurations in the
GS (ES) geometry
90