Document

This electronic excitation
transfer,
whose
practical description was first given by Förster,* arises
of a transition
depends
the
This electronic
excitationdipole
transfer, interaction,
whose practicalwhich
description
was firston
given
by magnitude
Förster,* arisesof the donor and acceptor
from a dipole-dipole interaction between the electronic states of the donor and the acceptor,
and
from transition
a dipole-dipole
interaction
between the
electronic
states of the donor
and the acceptor,
matrix
elements,
and
the alignment
and separation
of and
the dipoles. The sharp 1/r6
does notofinvolve
emission
and occurs
re-absorption
of a light field. Transfer occurs when the
does not involve the emission and re-absorption
a light the
field.
Transfer
when the
!"#$%
dependence on distance is often
used inof spectroscopic
characterization
of the
donor with the
oscillations
induced
electronic
on proximity
the donor areofresonant
oscillations of an optically induced electronic
coherence an
onoptically
the donor
are resonant
withcoherence
the
gap of thedepends
acceptor.
strength of the interaction depends on the magnitude
and energy
acceptor.
electronic
gap of the acceptor. The electronic
strength ofenergy
the interaction
on The
the magnitude
of a transition
dipole interaction,
whichand
depends
on the magnitude of the donor and acceptor
of a transition To
dipole
interaction,
whichthere
depends
the magnitude
the donor
acceptor
describe
FRET,
areonfour
electronicof states
that must
be considered: The electronic
6
transition
matrix elements,
the alignment
transition matrix elements, and the alignment
and separation
of the and
dipoles.
The sharp and
1/r6 separation of the dipoles. The sharp 1/r
ground and excited states of the donor and acceptor. We consider the case in which we have
!"#$%&#'(&$")*)+&',)&#-.'/#*)$0&#'1!(,/2'
dependencecharacterization
on distance is often
in spectroscopic
dependence
on distance is often used in spectroscopic
of theused
proximity
of donor characterization of the proximity of donor
&%'()*+,(-.,/'0%/0/%()1-)230%1-%,44-')5-0/,%,6,7(0/-4)%1)%8/)%.06,406)%10/)'0(,%9:;%)1%
and acceptor.and the acceptor is in the ground state. Absorption of light
excited
the donor electronic transition,
and acceptor.
8/)%.06,406)%)44,70(,%9<;%
Tothe
describe
FRET,
four electronic
states that
must
considered:by
Therapid
electronic
by
donor
at there
the are
equilibrium
energy
gap
is befollowed
vibrational relaxation which
To describe FRET, there are four electronic states that must be considered: The electronic
and We
excited
statesthe
of case
the donor
and we
acceptor.
ground&%=%*')'0%1,*4(-70%-/%',(.-/-%>()24-%>,(%6)%>(-.)%306')%1)%!0(*',(%40.,%-/',()5-0/,%
and excited states of the donor andground
acceptor.
consider
in which
have We consider the case in which we have
dissipates the reorganization energy
the electronic
donor λtransition,
of picoseconds. This leaves
D over the
excited
theofdonor
andcourse
the of
acceptor
excited1->060&1->060%'()%?6-%*')2%,6,7(0/-4-%1-%:%,1%<@%
the donor electronic transition, and the
acceptor
is in the
ground state. Absorption
light is in the ground state. Absorption of light
byoscillates
donor atbythe
equilibrium
energy
gap iswhich
followed by rapid vibrational relaxation which
the
donor
inequilibrium
a coherence
that
&!donor
-6%'()*+,(-.,/'0%)33-,/,%A8)/10%6,%0*4-66)5-0/-%1-%8/)%40,(,/5)%,6,7(0/-4)%
by the
at the
energy
gap
isthefollowed
rapid
vibrational
relaxation
12-2
dissipates
the reorganization
energy of theThis
donor
λD over the course of picoseconds. This leaves
0B4).,/',%-/107)%-/%:%=%(-*0/)/',%40/%8/%?)>%,6,7(0/-40%1-%<@%
λD over
the course of picoseconds.
leaves
dissipates
the
reorganization
the donor
at the
energy
gap energy
in theofdonor
excited
the donor in a coherence
that oscillates
D
the donor in a coherence that oscillates
state
q =
d D ) .FThe
ctronic energy
gapωof
acceptor
ω time-scale
. for gap in the donor excited
( q Aat=the0 )energy
eg (the
at the energy gap
inDthe donor
A
eg
excited
state so
ωegD this
( qD = d D ) . The time-scale for
ωegD ( qD =isd Dtypically
time-scale
for acceptor
staterelaxation
) . Thesubsequent
vibrational
and
FRET
nanoseconds,
&!is typically
-/%>()24)%6C,D4-,/5)%1,6%!"#$%*-%.-*8()%%
is typically
nanoseconds,MIT
so Department
this
FRETpreparation
nanoseconds,
this FRETof
Andrei Tokmakoff,
of Chemistry, 3/25/08
donor fluorescence.
In practice,
step the
is soefficiency
typically
much
%%%%%40/+(0/')/10%6,%,.-**-0/-%1-%E80(,*4,/5)%%
preparation
step is typically much preparation step is typically much
orescence emitted
acceptor.
fasterfrom
thandonor
the and
transfer
phase.
faster %%%%%1-%:%,1%<%
than the transfer phase.
For
For
faster than the transfer phase.
For
12.1.
FÖRSTER
coupled,resonance
weresonance
canenergy
writetransfer
our Hamiltonian
for
this energy
energy
require
a
resonance
transfer
we require a
wetransfer
require a we
resonance
condition,condition,
so that
tion theory
resonance
oscillation
of
the
oscillation
*
A + A* D
excited
of
the
RESONANCE ENERGY TRANSFER
condition, so that the
the
so resonance
that the
Förster
resonance energy
donor
oscillation
of
the
excited
transfer (FRET) refers to the nonradiative tr
donor
excited donor
excitation from a donor molecule to an acceptor molecule:
*(12.2)
Th. Förster, “Experimentelle
und theoretische
Untersuchung
des zwischenmolecularen Uebergangs von
theoretische
Untersuchung
des zwischenmolecularen
Uebergangs
von
Th. Förster, “Experimentelle und
*
*
Z. Naturforsch,
4a, 321 und
(1949); “Zwischenmoleculare Energiewanderung und
Electronenanregungsenergie,”
Z. Naturforsch, 4a, Electronenanregungsenergie,”
321 (1949); “Zwischenmoleculare
Energiewanderung
*
*
Th. Förster, “Experimentelle und
theoretische
Untersuchung
des zwischenmolecularen
Uebergangs
von Faraday
Fluoreszenz,”
Physik
2, 55 (1948);Discussions
“Transfer
Mechanisms
of Electronic Excitation,”
Discussions
Fluoreszenz,”
ofAnn.
Electronic
Excitation,”
Faraday
H
A A D Ann. Physik 2, 55 (1948); “Transfer Mechanisms
Soc. 27,Electronenanregungsenergie,”
7 (1959).
Soc.
27,
7
(1959).
Z. Naturforsch, 4a, 321 (1949); “Zwischenmoleculare Energiewanderung und
D + A→ D+ A
This electronic
excitation
transfer,
whose
practical
description
Fluoreszenz,” Ann. Physik 2, 55 (1948); “Transfer
Mechanisms
of Electronic
Excitation,”
Discussions
Faraday
was first g
ˆ
3 coupled,
⋅ rˆ )can
− μ Acould
⋅μ D our
onor and
acceptoracceptor
are weakly
write
Hamiltonian
for(12.2)
this
A ⋅ r ) μ Dwe
which
be and
moresubsequent
properly
written(12.3)
d * nD* a nA . The
= molecules,
,
nergydonor
to theand
acceptor Vleads
to vibrational
relaxation
acceptor
3
r
* perturbation
* and
*
hat can be solved
theory
donor
form
of a dipole-dipole
interaction:
H 0 between
Dby
A Hfrom
A +donor
Aacceptor
D fluorescence.
H A takes
A* D the In
D D the
hat isinteraction
spectrally
shifted
practice,
the efficiency
of
the distance
between
donor
and
acceptor
dipoles
and
is
a
unit
vector
that
marks
the
r̂
=
H H0 + V
!"#$%
ˆ
ˆ
3
⋅
μ
⋅
−
μ
⋅μ
r
r
)
)
A
D
A and
D acceptor.
r is obtained by comparing the fluorescence
emitted
from
donor
V =are
, electronic states
(12.3)
Hamiltonian
of The
the system
with the donor
excited,
is the
Hamiltonian
etween them.
dipole operators
here
taken and
to ronly
on the
3H A act
(12.2)
*acceptor are weakly coupled, we can write our Hamiltonian for this
he
donor and
* represents
* the electronic
excited.
D
and
nuclear configuration for
dependent
of
nuclear
i.e.
Condon
Weaboth
writevector
the that marks the
H 0r isA
D
H configuration,
A + A* Ddonor
H Athe
A* D
D D between
where
theAdistance
and
acceptorapproximation.
dipoles and r̂ is
unit
*
orm
can be solved
perturbation
theory
tor that
molecules,
which bycould
be more
properlyand
written
delectronic
nD* a nA . states
The for the
dipole
matrix
elements
that
couple
the
ground
excited
direction between them. The dipole operators here are taken
to only act on
the electronic states
:%,1%<%*0/0%)440>>-)2%1,G06.,/',%'().-',%H@%I-%>8J%)>>6-4)(,%'K@%1,66,%>,('8(G)5-0/-%
miltonian
of
the
system
with
the
donor
excited,
and
H
is
the
Hamiltonian
=
H
H
+
V
A
en
donor and
acceptor 0takes the form of a dipole-dipole interaction:
acceptor
as
and
be
* independent of nuclear configuration, i.e. the Condon approximation. We write the
(12.2)
and nuclear configuration for both
excited. D A representsˆ the electronic
3 A ⋅ r ) μ D ⋅ rˆ ) − *μ A ⋅μ D*
μ AA
(12.4) states for the
*
transitionHwhich
dipole
matrix
elements
that
and
excited
electronic
V = Dμ*could
, μ A*A
(12.3)
* * A +
* A couple
* Athe ground
A=
Awritten
3 more
d
n
a
n
.
The
or =
molecules,
be
properly
A
H
D
A
+
A
D
H
D
*
A
0
D r
A
D
donor
and acceptor
asthe form of *a dipole-dipole
donor
and acceptor
interaction:
μ
D μ DD* dipoles
D + and
D* μr̂ D*isD aDunit
ance
between
donor takes
and
vector that marks the (12.5)
D = acceptor
he Hamiltonian of the system with the donor excited,* and H
* A is the Hamiltonian
μ
=
μ
A
+
A
μ A* A A
(12.4)
*
ˆ )are
A⋅μ D to only
μ A ⋅ rˆ )( μhere
− μ Ataken
(
AA -/',()5-0/,%:&<%1,*4(-7)%40.,%-/',()5-0/,%1->060&1->060%
them.
The
dipole
operators
act
on
the
electronic
states
D ⋅ rseparate
* 3
the
dipole
operator,
we
can
the
scalar
and
, and nuclear configuration
(12.3)
for both
ptor excited. VD=A represents3 the electronic
r
ent
of nuclear
configuration,
i.e. be
the
* Dwrite the
μ Dmore
=Condon
D properly
μ DDapproximation.
D* written
+ D* μ Dd*We
(12.5)
*
al contributions
as which could
ceptor
molecules,
The
D nD* a n A .
nce
between
donor
andcouple
acceptor
dipoles
and
is a unitelectronic
vector that
marks
the
r̂excited
matrix
elements
that
the
ground
and
states
for
the
ˆ
ween donor and
the formwe
of acan
dipole-dipole
interaction:
μ Aacceptor
=
μ A takes
(12.6) the
For
theu A dipole
operator,
separate
scalar and
orhem.
as The dipole operators here are taken to only act on the electronic states
ˆ μ D ⋅in
3 interaction
μ A (12.3)
⋅μ D to be
rˆ ) −eq.
contributions
ws theorientational
transition dipole
A ⋅ r ) as
nt of nuclear configuration,
i.e.
the
Condon
approximation.
We write the(12.3)
V=
,
μ A = A μ AA* A* += ruˆA3* μμ A* A A
(12.6) (12.4)
A
A
A
0>,()'0(-%1-%1->060%-/1->,/1,/2%1)66)%40/L?8()5-0/,%/846,)(,%
atrix elements that couple the ground and excited electronic
states for the
distance κbetween donor and acceptor
dipoles
and
is
a
unit
vector
that marks the
r̂
*
*
9M0/10/%<>>(0N-.)20/;%
μ
=
D
μ
D
+
D
μ
D
This
allows
the
transition
dipole
interaction
in
eq.
(12.3)
to
be (12.5)
*
*
*
*
*
*
D
as
⎤ DD
D ATokmakoff,
(12.7)
=μ Aμ B 3 ⎡⎣ D A A D DD
+ A D Andrei
⎦ taken MIT
Department
of Chemistry,
3/25/08 states
12-3
een them.
The dipole operators here are
to only
act on
the electronic
r
written
as
*
*
ipole operator,
μ A =we can
μ AA*separate
A + Atheμ scalar
A and
(12.4)
A* A Condon approximation. We write the
endent
of
nuclear
configuration,
i.e.
the
κ
:
orientational factors are now
in
the
term
ributions as V =μ μ κ ⎡ D* * A A** D + A* D D* A ⎤
=
κ(12.7)
3 ( uˆ A ⋅ rˆfor
(12.8)
)( uˆ D ⋅ rˆ ) − uˆ A ⋅ uˆ D .
⎣D + the
⎦ electronic
ole matrix elements
states
μ D= DAthat
μBDDr*3couple
D ground
μ D* D Dand excited
(12.5) the
= uˆ A μ A
(12.6)
We can now obtain the rates of energy transfer using Fermi’s Golden Rule expressed as a
eptor All
as Aof the
orientational
factors
now in
the term κ:
pole operator, we
can separate
theare
scalar
and
transition dipole interaction in eq.
(12.3)
to
be in the interaction Hamiltonian:
*
*
correlation
function
μ
=
μ
A
+
A
μ
A
(12.4)
A
butions as
AA*
A* A
2π
1 +∞
2
0
function
inthe
the
interaction
Hamiltonian:
unction
interaction
Hamiltonian:
assumption
that
weinteraction
have
an equilibrium
system, even though we are initially in the excited donor
nction
inin
the
interaction
Hamiltonian:
ation
function
in
the
Hamiltonian:
correlation
function
inthere
the interaction
Hamiltonian:
.kmakoff,
ThisMIT
is Department
reasonable
for
the
case
that
is a clear time
scale separation between12-3
the ps
of
Chemistry,
3/25/08
+∞
2
π
1
2
+∞
2
π
1
2
state. w
This
for
is
scale separation
between the ps
1that+∞there
2 =
+∞at(clear
wk k 2πis 2reasonable
V
t2V
VI 0I()0(time
(12.9)
0)and
δ( (ωthe
ω
−case
ω
2ppthermalization
π VV=
1dt
)
)
)
(12.9)
dt
V
=
δ
ω
+∞
)
)
∑
kδ
k2−
I() V
2and
2∫π∫dt
2
1
ational=
relaxation
in
the
donor
excited
state
the
time-scale
(or
inverse
∑
k
k
I
w
p
V
V
t
(12.9)
ω
−
ω
(
)
(
(
2
−∞
∑k 2 k∑ p=
k
I VI ( t ) VI ( 0 )
Vk k =
(12.9)
δ ( ωwk −2ω∫−∞)−∞ 2pI∫ V dt=
2w
(12.9)
dt
V
t
δ
ω
−
ω
(
)
(
)VI(or
( 0 )inverse
∑
−∞
k
k
k
I
2
2
∫
vibrational relaxation and thermalization in the donor excited state and the−∞time-scale
!"#$%
of the energy transfer process.
ˆ we
uˆ D ⋅ are
rˆ ) −using
uˆ A ⋅ uˆ Da. correlation function there is an
=
κ 3 uˆSince
(12.8)
A ⋅ r ) we
his
is not
not
a
Fourier
transform!
isisrate)
isnot
a
Fourier
transform!
Since
are
using
a
correlation
function
there
is
an
a is
Fourier
transform!
Since
weSince
are using
a using
correlation
functionfunction
there is there
an is an
thenot
energy
transfer
process.
that thisof
a Note
Fourier
transform!
a correlation
* we are
*
that
this
is
not
a
Fourier
transform!
Since
we
are
using
a correlation function there is an
=transfer
Dthough
A and
kin=
Aexcited
D expressed
Now
substituting
therates
initial
state
theare
final
state
, wedonor
find
We
can
now
obtain
the
of
energy
using
Fermi’s
Golden
Rule
as a
that
we
have
an
equilibrium
system,
even
though
we
are
initially
the
excited
donor
that
we
have
an
equilibrium
system,
even
we
initially
in
the
* are initially in the excited donor
*
at we have
an have
equilibrium
system,
even
though
we
mption
thatNow
we
an
equilibrium
system,
even
though
we
are
initially
in
the
excited
donor
=
D
A
k
=
A
D
and
the
final
state
,
we
find in the excited donor
substituting
the
initial
state
assumption that we have an equilibrium system, even though we are initially
2
ion
functionfor
infor
the
interaction
Hamiltonian:
is reasonable
reasonable
for
the
case
that
aclear
clear
time
scale
separation
between
the
isreasonable
case
that
there
aclear
time
scale
separation
between
the
thethe
case
that
there
isisthere
ais
time
scale
separation
between
the
pspspsthe ps
κthere
+∞
1 the
This is reasonable
for
case
that
is
a
clear
time
scale
separation
between
* 2
*
case
there
is a clear time scale
separation between the ps
=
wETstate. 2 This
dtis reasonable
DκA μ Dfor
μ A (+∞
t ) μ Dthat
(12.10)
( t )the
( 0the
) μthe
A (0) D A
+∞
6
∫
2
π
1
−∞ 1in
2
*
*
relaxation
and
thermalization
in
the
donor
excited
state
and
time-scale
(or
inverse
elaxation
and
thermalization
the
donor
excited
state
and
time-scale
(or
inverse
r δdonor
axation
and=
thermalization
excited
state
(or
wET p inV2kthe
dt
D
Aμ
μ AV
t()tμ)time-scale
A (or (12.9)
(12.10)
( t dt
)and
(the
) )μ A time-scale
( 0excited
) Dinverse
wk and
V
0the
ωk the
−6and
ωdonor
ional=
relaxation
thermalization
excited
state
and
( in
)
(
D I( 0
∫=
I in
2 ∑
2 D
∫
−∞
vibrational
relaxation
thermalization
the
donor
stateinverse
and the time-scale (or inverse
−∞
r
− iH D t
energy
process.
D tprocess.
energy
ergy
μ Dtransfer
ttransfer
μ D eprocess.
. Here, we have neglected the rotational motion of the dipoles.
re
(transfer
) = eiHprocess.
of
the
energy
transfer
iH D t of the−energy
iH D t
rate)
transfer
+8/5-0/,%1-%
e
μtransform!
. * Since
weprocess.
have
neglected
the
motion
the dipoles.
atwhere
this is μnot
Fourier
are
using
a correlation
function
there of
is an
*Here, we
* * rotational
D ( ta) =
D e
*
*
= DD
k==
ADD, we
and
the
final
state
we
find
wubstituting
substituting
the
initial
state=average
*')'0%-/-5-)6,O%%%%%%%%%%%%%%%%%%%P%*')'0%L/)6,O%
=D
and
the
final
state
, ,we
find
the
initial
state
*the
*find
A AA
k k=state
AAD
the
initial
and
final
state
t substituting
generally,
the
orientational
is
=
D
A
k
Now substituting
thestate
initial
state
and
the
final
* = A D , we find
* 40((,6)5-0/,%1-%:%-/%
=
D
A
k
=
A
D , we find
and
the
final
state
Now
substituting
the
initial
state
tion
that
we
have
an
equilibrium
system,
even
though
we
are
initially
in
the
excited
donor
Most generally, the orientational average
is
48-%60%*')'0%-/-5-)6,%
2
2 22
κ
=
κ
t
κ
0
.
(12.11)
)
)
2
+∞ κ κκ
+∞ +∞
1
1
1
*
*
* that there is2 a clear time
* κ
* * separation between the ps
This iswreasonable
fordt
2 0scale
1dtthe+∞6 6case
=%!:R"%FS%+8/5-0/,%
=%
μ(D+∞0Dκ()0(μ0)t μ
μ
(12.10)
0)D)D
wwET 2 ∫22∫∫dt
DD
AμADμμ(DtD*()t(μt) )Aμμ(AκtA()t(1μ)t D)μ
(12.10)
(
(
D
A
AA0A D* A
(12.10)
0
=
κ
.*D
(12.11)
(
)
)
κ
A
A
ET ET
A
6
=
wET
dt
D
A
μ
t
μ
t
μ
μ
(12.10)
0
−∞
(
)
(
)
(
)
(
)
*
−∞ −∞
D
A
D
A
r
r
2r ∫−∞ =
6
wETifdonor
dt are
D
A μthe
(12.10)
( 0 ) μinverse
( 0 ) DtoA 1-%40((,6)5-0/,%>,(%
ever,
this factor
easier
to revaluate
the2 dipoles
static,
or( ttime-scale
if) μthey
rapidly
rotate
D
A (t ) μ
D (or
A
nal
relaxation
and is
thermalization
in the
state
and
6
∫excited
−∞
r
−D−
iH
t factor
t
t
this
iswe
easier
toneglected
evaluate
ifthe
the
dipoles motion
are
static,
orthe
ifdipoles.
they
rapidly rotate
to
− iH
tiHD D
μ
e
.Here,
we
have
neglected
rotational
motion
ofof
dipoles.
6C,.-**-0/,%1-%:%
=iHeenergy
eDiHtiHtDDt μ
μ
.−iHHere,
Here,
we
have
neglected
the
rotational
motion
the
dipoles.
iH
=tthe
e
.
have
the
rotational
of
the
)me
)μe=However,
2
D te
Dt
D
D
D
transfer
process.
=
e
μ
e
.
Here,
we
have
neglected
the
rotational
motion
of
the
dipoles.
isotropically
distributed.
For
the
static
case
κ
=
0
.
475
.
For
the
case
of
fast
loss
of
(
)
D
D
Dt
μ D ( t ) = eiH DtFor
μ Dthe
e−iHstatic
. case
Here, we
neglected
thecase
rotational
where
103,O%
% of of the dipoles.
become
isotropically
distributed.
κ 2 have
=
0.475
. For the
of fast motion
loss
*
*
lly,
the
orientational
average
is
y,
thethe
orientational
average
is
ally,
orientational
average
is =is
Now
substituting
the
initial
state
2 D A and the final state k = A D , we find
generally,
average
κ 2the→orientational
KMost
t
K
0
=κ
=2 32.
ntation:
(
)
(
)
generally,
the
orientational
average is
2 22
2
κ
=
κ
t
κ
0
.
(12.11)
κ
=
κ
t
κ
0
(12.11)
0κ()κ2)=
orientation: κ → K ( t ) κ2K (=
t)=
(12.11)
)(κκ) 0)t.=
)) κ2. 30.) /,6%4)*0%1-%(0')5-0/-%.06'0%3,604-%
.
(12.11)
2
κ
+∞
1
*
+8/5-0/,%1-%(12.11)
*
*t ) κ A0 )nuclear
κ
=
κ
.
A
or
D
,
and
the
D
and
coordinates
Since
the
dipole
operators
act
only
on
=
wET
dt 6 D A μ D ( t ) μ A ( t ) μ D ( 0 ) μ* A ( 0 ) D A
(12.10)
2
∫
−∞
sthis
is iseasier
tototo
evaluate
if ifthe
dipoles
are
static,
if,ifthey
rapidly
rotate
tototo coordinates
hisfactor
factor
evaluate
dipoles
are
oror
they
rapidly
rotate
factor
is easier
easier
evaluate
ifthe
the
are
static,
if
they
rotate
Astatic,
orareor
Dstatic,
and
D and
A
nuclear
the
operators
act
only
ver,
thisSince
factor
is dipole
easier
to revaluate
ifdipoles
the on
dipoles
orthe
if rapidly
they
rapidly
rotate to 40((,6)5-0/,%1-%<%-/%
this factor
is easier
to evaluate if the dipoles are static, or if they
rapidly rotate to
rthogonal,
canHowever,
separate
terms
in the donor
2 2 2and acceptor states.
iH Dwe
t 6,%400(1-/)',%1-%<%,%:%*0/0%*,>)()G-6-Q%A8-/1-O%
−
iH D t
48-%60%*')'0%-/-5-)6,%
pically
distributed.
For
the
static
case
κ
=
0
.
475
.
For
the
case
of
fast
loss
of
ropically
distributed.
For
the
static
case
κ
=
0
.
475
.
For
the
case
of
fast
loss
of
2
μtropically
= e distributed.
μ Ddistributed.
ewe canFor
. separate
Here,
we
have
the
motion
of the
the
static
case
κ donor
=
.acceptor
For .the
case
fast
lossdipoles.
of
( t )orthogonal,
Dare
terms
in neglected
the
and
me
isotropically
For
the
static
case
κ0.475
=
0rotational
.475
Forstates.
the of
case
2 of fast loss of
2
become
isotropically
distributed. For the static case κ =
0.475 . For the =%!<"%FS%+8/5-0/,%1-%
case of fast loss of
κ
+∞
1
2 22
* 2
*
2
2
2=
enerally,
wK K0 0 ==
dt
μD (t ) μD ( 0) D A μ A (t ) μ A ( 0) A
23. κ
κ κκ→
κ average
==
2+∞
32.3isD
40((,6)5-0/,%>,(%
→Kthe
Kt ()(torientational
κ
=
κ
=
.
62
1
*
*
→t))ETK(K((t)()w0)2) ∫K−∞
0
=
κ
=
2
3
.
ation: →κ 2(K=
(
)
r
2
2
D μKD ( t0) μ D=( 0κ) D=2 A
(12.12) 6C)**0(G-.,/'0%1-%<%
ET
2 ∫−∞
6 K (t )
κ2 κdt2 →r=
3 . μ A ( t ) μ A ( 0 ) A (12.11)
orientation:
(
)
κ
t
κ
0
.
)
)
*
*
κ
(12.12)
+∞only
1 act
A AAor2ororDDD,* and
the
DDD
and
AAnuclear
coordinates
dipole
operators
act
only
onon
, ,and
and
nuclear
coordinates
ehethe
dipole
operators
* the
and
the
and
A
nuclear
coordinates
ce
the
dipole
operators
act
only
on
%
C
t
C
t
=operators
dt
A
or
D
,
and
the
D
and
A
nuclear
coordinates
Since
the dipole
act
only
on
)
)
(
(
+∞
AA
2 ∫−∞ 1
6
*
D*κD*
r the
or Drapidly
, and rotate
the D to
and A nuclear coordinates
Since
act only
er, this factor is easier =to
evaluate
if6 the
dipoles are
static,onorAif they
CDoperators
dt dipole
* D* (t ) C AA (t )
2 ∫−∞
,
we
can
separate
terms
in
the
donor
and
acceptor
states.
nal,
separate terms
states.
r and
nal,we
wecan
can
termsin
inthe
thedonor
donor
andacceptor
acceptor
states.states.
thogonal,
weseparate
can separate
terms
in
the
donor
and acceptor
2
terms
in
this
equation
represent
the
dipole
correlation
forthe
the
donor
isotropically distributed.
For the
case κterms
=
0function
.in
475
case
ofinitiating
faststates.
loss in
ofthe
are orthogonal,
westatic
can separate
the. For
donor
and
acceptor
hat D
represents the electronic and nuclear configuration d nD* , we can use
Remembering that D* represents the electronic and nuclear configuration d * nD* , we can use
harmonic
oscillator
or energy gap Hamiltonian to evaluate 12-4
the
MIT
Department
of Chemistry,Hamiltonian
3/25/08
displaced
harmonic
oscillator
tions. For the case ofthe
Gaussian
statistics
we can
write Hamiltonian or energy gap Hamiltonian to evaluate the
functions.
For⎞ the
case of Gaussian
we can write
⎛and nuclear
*
hat D representscorrelation
the electronic
configuration
d * nDstatistics
* , we can use
2 − i ⎜ ω * − 2 λ D ⎟ t − g D (t )
⎠
CD* D* ( t ) = μ DD* e ⎝ DD
(12.13)⎞ *
⎛
2 − i ⎜ ω * − 2 λ D ⎟ t − g D (t )
⎝ DD
⎠
harmonic oscillator Hamiltonian or energy gapC Hamiltonian
the
( t ) = μ DD* toe evaluate
D* D*
*
2
−iω
*t−gA t
!"#$%
(12.13)
ctions. For the caseCof
Gaussian statistics
e AA we can. write
(12.14)
AA ( t ) = μ AA*
2 −iω * t − g t
A
AA
C
t
=
μ
e
.
(12.14)
(
)
⎛
⎞
*
*
AA
AA
2 − i ⎜ ω * − 2 λ D ⎟ t − g D (t )
use of
⎠
CD* D* ( t ) = μ DD* e ⎝ DD
(12.13)
Here we made use of
=ω * − 2λ D ,
(12.15)
D* D
DD
2 −iω * t − g (t )
A
ω * =ω * − 2λ(12.14)
(12.15)
C AA ( t ) = μ AA* e AA
.
D,
D D
DD
relative to the donor
s the emission frequency as a frequency shift of 2
relative to the donor
which expresses the emission frequency as a frequency shift of 2
use of
uency.
=frequency.
ω * − 2λ D ,
(12.15)
6,%+8/5-0/-%1-%40((,6)5-0/,%*-%>0**0/0%,*>(-.,(,%40.,%!$%-/3,(*,%1,66,%+8/5-0/-%1-%+0(.)%
absorption
D* can
D
DD expressed in terms of the inverse Fourier
ole correlation
functions
be
1-%G)/1)O% The dipole correlation functions can be expressed in terms of the inverse Fourier
fluorescence
or absorption
relative to the donor
es
the emission
frequencylineshape:
as a frequency shift of 2
transforms +∞
of a fluorescence or absorption lineshape:
1
D
− iωt
CD* D* ( t ) =
d
ω
e
σ
(12.16)
fluor ( ω )
1 +∞
D
− iωt
2π ∫−∞
d
ω
e
σ
(12.16)
* * (t )
pole correlation functions can be expressed in Cterms
of
the
inverse
Fourier
fluor ( ω )
∫−∞
D D
π
2
1 +∞
A
fluorescence or Cabsorption
lineshape:
(12.17)
t
=
d ω e− iωt σabs
(
)
( ω) .
AA
∫
1 +∞
A
− iωt
2π −∞
(12.17)
C
t
d
ω
e
σ
(
)
( ω) .
AA
abs
∫
+∞
1
−∞
D
− iωt
2
π
CD* D* ( t )in terms
d ωitse common
σ fluor ( ω
)
rate of energy transfer
of
practical
form, we make (12.16)
use of
2π ∫−∞
To express the rate of energy transfer in terms of its common practical form, we make use of
rem, which states that if a Fourier transform pair is defined for two functions, the
1 +∞
A
− iωt
Parsival’s
Theorem,
which
a Fourier transform
pair is defined for two functions, the
(12.17)
C
t
d
ω
e
σ
(
)
( ωstates
) . thatinifthe
AA
abs
∫
product of those functions2πis equal
whether evaluated
time or frequency
−∞
integral over a product of those functions is equal whether evaluated in the time or frequency
rate of energy transfer in terms of its common practical form, we make use of
domain:
∞
∞
orem, which states that if
* a Fourier transform
* pair is defined for two functions, the
uency.
* * ( )
fluor ( )
Dof
Dfrequency.
absorption
transforms
a fluorescence
2π −∞or absorption lineshape:
∫
D* D
=ω
DD*
− 2λ D ,
(12.15)
The dipole correlation functions
be −expressed
in terms of the inverse Fourier
+∞
1 can
iωt
D
1 which+∞ expresses
CD* D* −(itthe
d
ω
e
σ
ω
relative to(12.16)
the donor
frequency
as
a
frequency
shift of 2
)ωt emission
(
)
A ∫
fluor
−∞
(12.17)
C AAofta fluorescenced ω
e σ2abs
ω .
π lineshape:
transforms
or absorption
()
∫
−∞
frequency.
2πabsorption
s the rate of energy transfer in
( )
!"#$%
1 1 +∞ − iωt −can
TheCdipole
in terms of the inverse
Fourier
.
(12.17)
t correlation dfunctions
ω e σiωAt beDωexpressed
+∞
C
d ω e absσ( fluor) ( ω)
AAD(* D)* ( t )
∫
−∞ ∫−∞practical form,
terms
of
its
common
we
2
π
π
2
transforms of a fluorescence or absorption lineshape:
make use of
(12.16)
1
To express
energy transfer
in terms
ofis(its
common
practical
form, we make
of
1 C +∞
Theorem, which
states the
thatrate
if aofFourier
transform
pair
for
two functions,
the use(12.17)
− iωt d
tdefined
(12.16)
)
C t
d ω e ∫ σ Aω eω .σ ( ω)
+∞
AA ( )
abs ( )
− iωt
2π
$,0(,.)%1-%T)(*-3)6%
2π ∫−∞transform pair is defined for two functions, the
Parsival’s Theorem, which states that if a Fourier
1 in
ver a product9*')G-6-*4,%
of those functions
is equal
whether
evaluated
the time
or frequency 3-*')%
,A8-3)6,/5)%
1-% 18,%
()>>(,*,/')5-0/-%
1,6%
C (t )
d ω e*,?/)6,%
σ ( ω) . 1)6% >8/'0% 1-% (12.17)
∫
To express
rate ofofenergy
transfer inisterms
its
practical
make use of
2π common
integral
over athe
product
those functions
equal of
whether
evaluated
in theform,
time we
or frequency
,/,(?,240;%
To express
the rate
ofifenergy
transfertransform
in terms of pair
its common
practical
make use the
of
Parsival’s Theorem, which
states
that
is defined
forform,
two we
functions,
domain:
fi (t)
/a f!Fourier
1,L/-')%8/)%40>>-)%1-%!08(-,(%%%%%%%%%%%%%%%%%%%%%%%%%>,(%18,%+8/5-0/-%f
i (! )
1%,1%f2Q%6C-/',?()6,%1,6%
Parsival’s
∞
∞ Theorem, which states that if a Fourier transform pair is defined for two functions, the
∞
integral over a *product of those
functions
is ∞equal whether evaluated in the time or frequency
>(01070%1,66,%18,%+8/5-0/-%=%8?8)6,%*-)%4K,%*-)%3)68')'0%/,6%10.-/-0%1,-%',.>-%4K,%
*of those
integral
over
a
product
whether evaluated in(12.18)
the time or frequency
*
t
f
t
dt
=
f
f
ω
dfunctions
ωf1 (. ) fis2* equal
(
)
(
)
t
f
t
dt
=
ω
d
ω
.
(12.18)
1( ) 2 ( )
1
2
(
)
(
)
(
)
∫
∫
/,6%10.-/-0%1,66,%+(,A8,/5,@%
1
2
∫
∫
domain:
domain:−∞
−∞
−∞
−∞
%
∞
∞
*
between
the donor
allows
us totransfer
express the
energy
transfer
rate
as
overlap
=f *∫J fω( integral
. the
(12.18)
( tintegral
) ff*an
((t ω
) dt
) fω* (.ω ) dωJDA
between
donor
ws us to express
energy
rate
as
an
overlap
% Thisthe
∫
f
t
f
t
dt
=
d
(12.18)
DA
(
)
(
)
)
(
)
2
∫−∞ 1 2
∫−∞ 1
% fluorescence and acceptor
absorption spectra:
This allows us to express the energy transfer rate as an overlap integral J between the donor
D* D*
−∞
+∞
AA
∞
A
abs
∞
1
2
1
−∞
2
−∞
ce and acceptor absorption spectra:
2
− iωt
−∞
D
fluor
DA
This allows us to express
the energy
transfer
ratespectra:
as an overlap integral JDA between the donor
2
fluorescence
acceptor
absorption
κand
2
2 +∞
1
A
D
2
w
d
(12.19)
=
μ
μ
ω
σ
ω
σ
2
(
)
*
*
κ
ET
abs
fluor ( ω ) .
fluorescence
and
acceptor
absorption
spectra:
+∞
6
2
2
DD A AA
1
2 +∞
1 κ∫−∞D 2
A
D
r
wET ( 2ω )6σμ DD (μω
d ω σ abs ( ω) σ fluor ( ω)(12.19)
.
(12.19)
wET
d2ω=
=
μ DD* μ AA*
σ
AA ) .∫−∞
abs
fluor
r
2
6
−∞
r lineshape normalized
2 +∞
1 κto the 2
Here σ is the
A
lineshape
normalized
to thed ω σ abs
Here
wETσ is the
(12.19)
=
μ
μ
ω) σ Dfluor ( ω) .
(
2
6
∫
DD*
AA*
−∞
r
the lineshapetransition
normalized
to the element
transition matrix
matrix
squared:element squared:
2
normalized
to the
Here σ2 is the lineshape
σ = σ / μ . The overlap integral is a
σ
=
σ
/
μ
.
The
overlap
integral
is
a
matrix element squared:
transition matrix element squared:
∫
.
-/',?()6,%1-%03,(6)>Q%U:<%
-/',?()6,%1-%03,(6)>
The overlapσ =integral
σ / μ . is
Thea overlap integral is a
2
*
*
yetween
transfer
rate
scales
as rtransitions.
, depends on
donor
acceptor
6 the strengths of the electronic
strengths
of and
the
electronic
o,
the
energy
transfer
rate
scales
as
r
, neglected
depends
on
the neglected
strengths
the the
electronic
ptor
absorption.
One absorption.
of the6 thingsOne
we of
have
ishave
that
the rate ofof
transfer
andfluorescence
acceptor
the
things we
isenergy
that
rate of energy transfer
een
donor
nd
acceptor
molecules,
and
requires
resonance
between
donor
fluorescence
transfer
rate
scales
as
r
,
depends
on
the
strengths
of
the
electronic
between
donor
fluorescence
s for donor
andrate
acceptor
molecules,
and requires
resonance
between
fluorescence
depend
on
the
of excited
donor
state
population
relaxation.
Sincedonor
this relaxation
is this relaxation is
will
also
depend
on
the
rate
of
excited
donor
state
population
relaxation.
Since
rate
of
energy
transfer
One
of
the
things
we
have
neglected
is
that
the
rate
of
energy
transfer
don.
acceptor
molecules,
and
requires
resonance
between
donor
fluorescence
t the absorption.
rate of energy
transfer
ptor
Onedonor
of thefluorescence
things we have
neglected
is that
the rate
of energy
transfer
dominated
by
the
rate,fluorescence
the rate of
energy
transfer
isenergy
commonly
!"#$%,%0**,(3)G-6-%*>,(-.,/')6-%
typically
dominated
by
the
donor
rate,
the
rate
of
transfer is commonly
Since
this
relaxation
is
e One
rateSince
of the
excited
donor
state
population
relaxation.
Since
this relaxation
is
.ion.
things
we
have
neglected
is
that
the
rate
of
energy
transfer
this
relaxation
is
depend on the rate of excited donor state population relaxation. Since this relaxation is
fluorescence
lifetime
of the donor
τ : of the donor τ :
terms of
effective
R0, and the
and
the
fluorescence
lifetime
written
indonor
termsdistance
of anpopulation
effective
distance
R0,Since
transfer
isancommonly
y theoftransfer
donor
fluorescence
rate,
the rate
of energy
transfer
is commonly
ergy
is
commonly
rate
excited
state
relaxation.
this
relaxation
is
dominated by the donor fluorescence rate, the rate of energy transfer is commonly
6
6
feffective
theofdonor
τfluorescence
: τR0:, and rate,
the fluorescence
of the
donor
τ :
distance
Rlifetime
1 of
⎛
⎞
the
donor
the
rate
energy
transfer
is
commonly
me
the
donor
0
R
1
⎛
⎞
fluorescence
lifetime
of the donor τ :(12.20)
n terms of an effective distance R
0
= the
w0ET, and
⎜
⎟
=
w
(12.20)
ET
⎜
⎟
τ
r
⎝
⎠
D
6
fluorescence
lifetime
of theτdonor
fective distance 4066,?).,/'0%)%0**,(3)G-6-%*>,(-.,/')6-%9+0(.86)%V>()24)C;O%
R0, and the
D ⎝ r ⎠τ :
6
R
1
⎛
⎞
0
(12.20)
1 ⎛ R0 ⎞
= the⎜ rate
wET
(12.20)
(12.20)
⎟
(or
probability)
of
energy
transferof
isenergy
equal to
the
rate
tical transfer
distance
R
=
w
(12.20)
0
6
ET
⎜
⎟
τ D ⎝ rdistance
the
rate
(or
probability)
transfer
is equal to the rate
At the critical transfer
R
⎠
0
τD ⎝ r ⎠
1 ⎛ R0 ⎞
wET = in ⎜terms
defined
of the in
sixth-root
ofthe
thesixth-root
terms in(12.20)
eq.
(12.19),
and
is (12.19), and is
scence.
Rfluorescence.
⎟ defined
0 is
is
terms
of
of
the
terms
in
eq.
of
R
fer
is
equal
to
the
rate
0
τ
r
the
rate
(or
probability)
of
energy
transfer
is
equal
to
the
rate
distance
R
transfer
is
equal
to
the
rate
⎝
⎠
0
itical transfer distance RD0 the rate (or probability) of energy transfer is equal to the rate
3,604-'Y%1-%#$%1->,/1,%)/4K,%1)6%
yinwritten
as
eq.incommonly
(12.19),
and
is
written
as
sscence.
defined
in
terms
of
the
sixth-root
of
the
terms
in
eq.
(12.19),
and
is
erms
(12.19),
and
is
the
rate
(or
probability)
of
energy
transfer
is
equal
to
the
rate
istance
Req.
0 0 is defined in terms of the sixth-root of the terms in eq.
(12.19),
and is
R
',.>0%1-%(-6)**).,/'0%1-%:RQ%1-%
2 ∞
D
9000 ln(10)φD κ
σ fluorφin
ε2A ((12.19),
ν ) D and is
(Dν )eq.
∞
9000
ln(10)
κ
of
the
terms
ydefined
writteninas termsR06of= the sixth-root
) ε A (ν ) (12.21)
d
ν
fluor (ν *06-'0%10.-/)'0%1)66)%1-/).-4)%1-%
5 4R 6 =
4
∫
(12.21)
d
ν
π
ν
n
N
128
0
4
∫
2 ∞
0 128π 5 n 4 N
D
E80(,*4,/5)%
2
ν
9000
ln(10)
φ
κ
D
∞
) 6
0
D9000 ln(10)φ fluorκ(ν ) ε A (ν )
D
fluor (ν ) ε A (ν )
6 (12.21)
(12.21)
(12.21)
R0 =
d
ν
(12.21)
R50 n=4 N 2 ∫ 40/'(-G8'0%V*063,/'%*4(,,/-/?C%
4 dν
5 D4
4
∫
π
ν
128
e practical
definition
which
accounts
for
the
frequency
dependence
of the transition∞
0 π n N(ν ) ε (ν )
ν
128
9000
ln(10)
φ
κ
0
D
This is the practical
definition
which
accounts for the frequency dependence of the transitionA
fluor
6
(12.21) in common
R
=
d
ν
0
eraction
and128
non-radiative
donor relaxation
in addition to being expressed
5 4
4
∫
103,O%
π
ν
n
N
pendence
oftransitionthe
transitioninteraction
and
non-radiative
donor
relaxation
in addition
being expressed in common
0
efinition
which
accounts
for
the frequency
dependence
of
the
transitionence
ofdipole
the
-1 the
D transitionhe
practical
definition
which
accounts
dependence
of to
the
represents
units of frequency
in cmfor
.&W%Thefrequency
fluorescence
spectrum
σ
luor must be
-1
n%-/1-4,%1-%(-+()5-0/,%1,6%.,550%
being
expressed
in
common
ν
represents
units
of
frequency
in
cm
.
The
fluorescence
spectrum σ Dluor must be
units.
non-radiative
donor
relaxation
in
addition
to
being
expressed
in
common
expressed
in
common
+(,A8,/5)%-/%4.
teraction
and
non-radiative
donor
relaxation
in
addition
to
being
expressed
in
common
D
finition
which
for fluor
the νfrequency
dependence
the transitiond to unit
area,accounts
in cmof(inverse
wavenumbers).
The
) is expressed
D so that
D
-1
D
D
N%/8.,(0%1-%<30?)1(0%
-1
D
ce
spectrum
σ
must
be
normalized
to
unit
area,
so
that
ν
is
expressed
in
cm
(inverse
The
units
of
frequency
in
cm
.
The
fluorescence
spectrum
σ
must
be
/0(.)6-55)')%)1%)(,)%8/-')(-)%
fluor
)
ectrum
σ
must
be
representsluordonor
units of frequency
in cm .toThe
σ luor mustwavenumbers).
be
fluor
luor
in addition
beingfluorescence
expressed
inspectrum
common
non-radiative
spectrum A D(ν ) relaxation
must be expressed
in molar
decadic
extinction
coefficient
units
D
#:The
%E80(,*4,/4,%6-+,2.,%1,6%10/)'0(,%
nverse
wavenumbers).
The
sounit
thatarea,
σ fluorso
νin)that
is
cm
wavenumbers).
absorption
ν)fluorescence
must
be (inverse
expressed
in(inverse
molar
decadic
extinction
-1expressed
D
(spectrum
,*>(,**)%-/%4.%
erea,
The
(
)
edwavenumbers).
toof
isinexpressed
in cm
wavenumbers).
Thecoefficient units
Aν
fluor
nits
frequency
cm
.
The
spectrum
σ
must
be
cm). n is the index of refraction of the solvent, N is Avagadro’s
luor number, and φD is the
extinction
coefficient
units
must
be
inexpressed
molarofdecadic
extinction
coefficient
units
(ν )coefficient
(liter/mol⋅cm).
n,*>(,**)%-/%6-'(-X.06!4.%
is
the
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ofdecadic
the solvent,
N isThe
Avagadro’s
and φD is the
ction
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εD A (expressed
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must
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units
)
a,
so
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expressed
in cm
wavenumbers).
)
orescence
quantum
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fluor
and
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thequantum
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edro’s
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of
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ofis(,*)%A8)/24)%1-%E80%1,6%10/)'0(,%
the
solvent, Nthe
is solvent,
Avagadro’s
and φD is theand φD is the
donor
fluorescence
⋅cm).
n
is
the
index
refraction
N is number,
Avagadro’s
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and
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D
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in molarof yield.
decadic
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A( )
antum
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orescence
quantumofyield.
ndex
of refraction
the solvent, N is Avagadro’s number, and φ is the
ntum yield.
D
d Förster theory (GFT), or indeed anytime that sums over
ded to find the total energy transfer rate.
er theory is that it connects Equation (4) to spectra that can be
mentalist. These expressions are well-known and the reader is
e for further explanation [1–4,60]. The Förster critical transfer
particularly useful. It is defined as the interchromophore distance
ansfer efficiency is 50%. Its application, however, relies on use of
ximation for the electronic coupling so that the energy transfer
!"#$O%40/1-5-0/-%1-%3)6-1-'Y%
"Z%%40((-*>0/1,%)66)%1-*')/5)%'()%-%18,%4(0.0+0(-%)66)%A8)6,%6C,D4-,/5)%1,6%'()/*+,(%=%[Z\O%
E¼
1
:
6
1 þ ðR=R0 Þ
ð5Þ
is formula to cases where the dipole–dipole approximation fails, a
cal distance$K,%>(-.)(]%)**8.>20/*%-/%!0(*',(%'K,0(]%)(,O%%
RG for the donor–acceptor pair has been proposed
9W;%'K,%,6,4'(0/-4%408>6-/?%-*%)>>(0N-.)',6]%'K,%1->06,^1->06,%408>6-/?P%%
ore precisely determined electronic coupling, V, to that estimated
s 2 !1 n 4
9_;%'K,%*063,/'%*4(,,/-/?%0+%'K-*%-/',()420/%-*%
formula (see
Equation (3b)), V dd, enables a scaling factor
und so that:9`;%0/6]%0/,%,6,4'(0/-4%*')',%0+%,)4K%0+%'K,%10/0(%)/1%'K,%)44,>'0(%)(,%0+%-/',(,*'P%%
9a;%'K,%,6,4'(0/-4%408>6-/?%-*%3,(]%b,)cQ%.,)/-/?%-'%-*%*.)66%40.>)(,1%'0%'K,%K0.0?,/,08*%
RG ¼ !R0 :
ð6Þ
*>,4'()6%6-/,%G(0)1,/-/?%90(%G)'K%(,0(?)/-*)20/%,/,(?];%)/1%%
ted for R0 in
Equation (5), which is useful if electronic couplings
9[;%-'%-*%)**8.,1%'K)'%'K,%E84'8)20/*%0+%'K,%,/3-(0/.,/'%)(08/1%,)4K%.06,486,%)(,%
ntum–chemical
calculations.
-/1,>,/1,/'@%%
ptions in %Förster theory are: (1) the electronic coupling is
e–dipole coupling, Equation (3b); (2) the solvent screening of this
M-)*48/)% 1-% A8,*',% ->0',*-% /,6% 40(*0% 1,6% ',.>0% =% *')')% )/)6-55)')% ,% -/% >)(',% *8>,()')%
) only one electronic state of each of the donor and the acceptor
>(0>0/,/10%
?,/,()6-55)5-0/-%
ctronic coupling
is very weak,
meaning it is small )66C)>>(044-0%
compared to the 1-% !0(*',(@% T,(% ,*,.>-0Q% A8)/10%
6C)>>(0**-.)5-0/,%9W;%4)1,%,%6C-/',()5-0/,%/0/%=%>-d%1,*4(-3-G-6,%*060%40.,%1->060&1->060Q%=%
line broadening
(or bath reorganisation energy) and (5) it is
1,L/-(,%
8/)%
/803)%
1-*')/5)%
4(-24)% "eF$"Z% 4K,% 2,/,% 40/'0% 1,66)% 1-3,(*)% /)'8()%
tions of the>0**-G-6,%
environment
around
each
molecule
are independent.
1,66,%-/',()5-0/-@%
previous reviews
[1,11] for discussion of assumptions (1), (2) and
that have now resolved the nature of solvent screening [62–64].
:,N',(%#$%
f/1->,/1,/',.,/',Q%:)3-1%g@%:,N',(%>(0>0*,%8/%)6'(0%.,44)/-*.0%4K,%>(,3,1,%8/0%
*4).G-0%1-%,6,7(0/-@%%
%
&%-/-5-)6.,/',%+0(.86)'0%>,(%1,*4(-3,(,%6C,/,(?]%'()/*+,(%-/%4(-*')66-%10>)2%40/%-0/-%
9+0*+0(-;%
&%2>-4).,/',%)**04-)'0%40/%'()*+,(-.,/2%4K,%40-/306?0/0%'()/*-5-0/-%1->060&>(0-G-',%
9,*,.>-0%'()*+,(-.,/'0%'(->6,70&'(->6,70;%
&%/,4,**)(-0%03,(6)>%'()%+8/5-0/-%1C0/1)%,6,7(0/-4K,%1-%:%,1%<%
&%6C,D4-,/5)%1,4)1,%,*>0/,/5-)6.,/',%40/%6)%1-*')/5)%
&%(-4K-,1,%1-*')/5,%.06,406)(-%>-d%40(',@%I-%>)(6)%)/4K,%1-%V*K0('&1-*')/4,%#$C%
&%?,/,()6.,/',%)**04-)'0%)6%A8,/4K-/?@%f6%',(.-/,%hA8,/4K-/?h%-/1-4)%A8)6*-)*-%>(04,**0%
L*-40%4K,%(-184,%6)%E80(,*4,/5)@%%
" 2r %
wDexter = KJ exp $ ! '
# L&
L%=%6,?)'0%)66)%*0..)%1,-%()??-%1-%H)/%1,(%i))6*%
!0(*',(%3*%:,N',(%
!0(*',(%3*%:,N',(%
?,/,()6-55)5-0/,%
e2
V=
4!" 0
"
PDD* (r1 )PA*A (r2 )
dr1dr2
r1 ! r2
%%%%%%%%%%%%%%%%%)/1%%%%%%%%%%%%%%%%%%%%)(,%'()/*-20/%1,/*-2,*@%%
PDD* (r1 )
PA*A (r2 )
International Reviews in Physical Chemistry
57
A8)/10%:%,1%<%*0/0%*,>)()2%1)%8/)%1-*')/5)%.)??-0(,%1,66,%60(0%1-.,/*-0/-%9*0..)%1,-%
second effect
considered by Dexter is that when donor and acceptor approach each
()??-%1-%3)/%1,(%i))6*;%)660()%H%>8J%,**,(,%)>>(0**-.)'0%40/%8/C-/',()5-0/,%1->060&1->060%
osely, their9)>>(0**-.)5-0/,%1-%!0(*',(;@%
molecular orbitals can overlap [65]. Even if this happens to a small
he correction
to the electronic coupling might be significant. Dexter considered the
<%1-*')/5,%.-/0(-Q%?6-%0(G-')6-%.06,406)(-%1-%:%,1%<%>0**0/0%*03()>>0(*-@%</4K,%8/)%>-4406)%
e correction
to the electrodynamical interactions, term 2 in Equation (2). Using the
*03()>>0*-5-0/,%>8J%)3,(,%,j,B%*-?/-L4)/2@%f/%A8,*'0%4)*0%*-%'(03)%6)%1->,/1,/5)%
,*>0/,/5-)6,%1)66)%1-*')/5)@%
n approximation
for two-electron integrals,
1
ðiuj jvÞ # Siu Sjv ½ðiij jj Þ þ ðiijvvÞ þ ðuuj jj Þ þ ðuujvvÞ&,
4
k866-c,/%)>>(0N@%+0(%'b0%,6,4'(0/*%
ð11Þ
-/',?()6%
e found that the Dexter exchange interaction scales with the product of orbital
D * A * AD ! SD*A*SAD 0 0
s of donor–acceptor orbitals:
ða b jbaÞ / Sa0 b0 Sab .
exp("r / L)
AD !
key result Sof
Dexter’s
work (which holds also for more rigorous treatments of
overlap, vide infra) is that owing to the way orbital tails decay, Sab / expð'!RÞ, the
1->,/1,/1,/5)%40.,%
,N>9&"Xg;%
In(calculated total electronic coupling)
Downloaded By: [Canadian Research Knowledge Network]
energy transfer rate, the mechanism of singlet–singlet energy transfer has been categorised
as Förster (rate / 1/R6) or Dexter (rate / exp['4!R]). These mechanisms coexist for
singlet–singlet energy transfer, as shown by Equation (2), because the total electronic
coupling is the sum V toal ¼ V ed þ V ioo. An example of the total electronic coupling, V toal,
!0(*',(%3*%:,N',(%
for the interaction between the lowest electronic state, S1, and the second electronic state
S2, of two naphthalene molecules is shown in Figure 2 [55]. Notice how the approximately
1/R3 distance dependence (cf. the dotted lines) seen in the distance range 5–10 Å changes to
a steeply rising exponential function when R 5 5 Å as V ioo becomes significant.
10
y
8
X
S2
6
z
R
4
S1
2
0
3
9
7
8
5
6
4
Intermolecular separation, R (Å)
10
1->,/1,/1,/5)%40.,%
`%
naphthalene
the S2 states as a
WX"and
Figure 2. Total electronic coupling between the S1 states of
function of interchromophore separation. The steep rise at short separations indicates the onset of
interactions depending on interchromophore orbital overlap.
Reproduced with permission from Ref. [55]. Copyright 1994, American Chemical Society.
!"#$%
International Reviews in Physical Chemistry
Vol. 30, No. 1, January–March 2011, 49–77
Energy transfer from Förster–Dexter theory to quantum coherent
light-harvesting
Alexandra Olaya-Castroa* and Gregory D. Scholesb
By: [Canadian Research Knowledge Network] At: 08:13 28 March 2011
a
Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, United Kingdom; bDepartment of Chemistry, Institute for Optical Sciences
and Centre for Quantum Information and Quantum Control, University of Toronto,
80 St. George Street, Toronto, ON M5S 3H6, Canada
(Received 10 August 2010; final version received 28 October 2010)
Electronic excitation energy transfer is ubiquitous in a variety of multichromophoric systems and has been a subject of numerous investigations in the
last century. Recently, sophisticated experimental and theoretical studies of
excited state dynamics have been developed with the purpose of attaining a more
detailed picture of the coherent and incoherent quantum dynamics relevant to
energy transfer processes in a variety of molecular aggregates. In particular, great
efforts have been made towards finding experimental signatures of coherent
superpositions of electronic states in some light-harvesting antenna complexes
and to understand their practical implications. This review intends to provide
some foundations, and perhaps inspirations, of new directions of research. In
particular, we emphasise current opinions of several effects that go beyond
normal Förster theory and highlight open problems in the description of energy
transfer beyond standard approximations as well as the need of new approaches
to characterise the ‘quantumness’ of excited states and energy transfer dynamics
in multichromophoric systems.
Keywords: electronic energy transfer; electronic coherence; FRET; Förster
theory; two-dimensional spectroscopy; light-harvesting