energetics of electron transfer...
TRANSCRIPT
Energetics of Electron Transfer Reactions
Dmitry MatyushovArizona State University
MIT, November 17, 2004
Problems
Tunneling between localized states
<∆E>
∆E
D A
E
HOMO
LUMO
Instantaneous energy gap ∆E becomes the reaction coordinate
Reaction Coordinate
Linear Relation
Marcus-Hush Theory of Electron Transfer
Two-parameters model:
∆F act =(λ + ∆F0)
2
4λ
λ is the reorganization energy
∆F0 is the driving force.
Fi(X) = F0i +(X − X0i)
2
4λ
2λ = X02 − X01
Energy gap law is the samefor charge separation (CS)
and charge recombination
(CR)
-2 0 2 4X
0
2
Fi(X
)
X=0
∆F0
<δX2>=2λkT
-2 0 2 4-∆F
0
-2
0
-∆F
act CS CR
Charge-Transfer Processes in Condensed Phase
Density of energy gaps (X = hω):
FCi(ω) = h〈δ (hω − ∆E(q)〉q,i = e−βFi(hω)
y
y
spectroscopy charge transfer
1 1.5 2 2.5
energy gap/frequency
0
0.5
1
1.5
2
FCi(X
)
<X>=νabs/em
ET kinetics
<δX2>=Γ
abs/em 2
optical spectroscopy
X=0
Spectral Band-Shape
15 20 25
ν/kK
0
0.2
0.4
0.6
0.8
Inte
nsity
solvent-induced
gas-phase
Convolution of gas-phase and solvent-induced bands:
FCi(ν) =
∫
FCi,solvent(x)FCi,gas(ν − x)dx
Inhomogeneous Solvent-Induced Broadening
Iabs/em(ν) =[
2πσ2abs/em
]−1/2exp
(
−(ν − νabs/em)2
2σ2abs/em
)
0 0.5 1 1.5 2 2.5 3
frequency/eV
0
0.5
1
1.5
2
Inte
nsity ∆ν
solv
νgas
Γabs/em
Rint
σ2abs/em =
Γ2abs/em
8 ln(2), λabs/em =
σ2abs/em
2kT.
What do we expect to see? Steady-state spectra.
Solvent component of the band:
0 0.5 1
h∆νst
0
0.5
1
σ2 /kT
solvent polar
ity
Consistency condition:
σ2abs
kBT=
σ2em
kBT= h∆νst
0 1 2 3 40
0.5
1
1.5∆ν
st
σabsσ
em
Total width vs the total Stokes shift:
0 0.5 1
h∆νst
0
0.5
1
σ2 /kT
λvν
v/kT
solvent polar
ity
What do we expect to see? Time-Resolved Spectra
Time-resolved excitation:
g
e t = 0
t = ∞
Stokes shift correlation function:
SΩ,i(t) =〈Ω(t)〉i − 〈Ω(∞)〉i〈Ω(0〉i − 〈Ω(∞)〉i
Equilibrium correlation function:
Ci(t) =〈δΩ(t)δΩ(0)〉i〈δΩ(0)2〉i
TRF broad-band excitation:
g
et = 0
t = ∞
Linear response (parabolic free
energy surfaces):
SΩ,i(t) = Ci(t)
λi(t) = λ = Const
Deviations from the Gaussian Picture.
Experimental Evidence:
• Asymmetry between CS and CR
energy gap laws
• Asymmetry between steady-
state absorption and emission
lines
• Change in the time-resolved op-
tical width
• Coumarin puzzle
4 4.4 4.8 5.2 5.6 6 6.4
∆−νst/kK
8
10
12
14
βσ2 /k
K
solvent polarity
absorption
emission
The Approximation of Fixed Charges
Charge transfer+
++
−−
−
−
−
−+++
+
+
−−m01 m02
ej(1)
ej(2)
Interaction(i) = −∑
j
e(i)j φ
(i)j
φ(i)j is the potential of the solvent at charge j.
The Approximation of Fixed Charges. Simulations
0 1 2 3 4 5 6 7y
0.2
0.3
0.4
0.5
βλi/(
q* )2
<(δu0s
)2>
0
<(δu0s
)2>
1
+ -1:
0:
Polarizable Solute
H(i) H(i)0 − 1
2
∑
j e(i)j φsolv
j − W(i)pol
−−−−−−−−−−−→dipolar solute
H(i)0 −m0i ·R−
1
2α0iR
2
y
α0i is the solute dipolar polarizability
Two sources of “polarizabil-
ity”:
• D-A coupling through
mDA
• Coupling of D and A states
to other electronic states
mDk
mAm
m DA
D
Ak
m
Q-Model
Hamiltonian:
Hi = H(i)0 −m0i·R−
1
2α0iR
2m
Dk
mAm
m DA
D
A
Free energy surfaces:
F (X) = F0 +
(√
|α|
∣
∣
∣
∣
X − ∆F0 +α2λ1
1 + α
∣
∣
∣
∣
− |α|√
λ1
)2
α =(
3
√
λ1/λ2 − 1)−1
When α → ∞, the parabolic surfaces are obtained
F (X) = F0 +(X − ∆F0 − λ)2
4λ, λ1 = λ2 = λ
Q-Model: Derivation
Q-Model: Properties
X
Fi(X
)
F2(X)=F1(X)+X
band boundary
2
1X0
different curvatures, λ1, λ2
linear asymptote
Connection to spectroscopic observables:
λi = βh2〈δν2〉i/2, α = −h∆νst + λ2
λ2 − λ1
∆F0 = hνabs+νem
2− λ1
2α
(1+α)2
Marcus-Hush term Correction from non-parabolicity
Reorganization Energy of Polarizable Chromophores
λi = ap (fi/fei)[
∆m0 +2ap ∆α0 m0i
]2
nuclear polarization response, µp = −apm20
dipole moment change polarizability change
∆α0 = fe2α02 − fe1α01, ∆m0 = fe2m02 − fe1m01
fei =[
1 − 2 ae α0i
]−1fi =
[
1 − 2apα0i
]−1
ae is the response of induced polarization, µe = −aem20
MC Simulations of Transitions in Polarizable Chromophores
dipole moment m
polarizability α
diameter σ
CS transition CR transition
dipole moment m0polarizability α0
Energy gap:
∆E = ∆E0−m0·R−1
2α0R
2
Reorganization energy:
λi = β〈[δ(∆E)]2〉i/2
0 0.05 0.1
α0/σ3
1
1.5
2
2.5
λ i, eV
Q-model
MC
JPCA, 108, 2004, 2087-2096.
0 20 40 60 80
βm0
2/σ3
0
5
10
15
20
-βµ e
slope=aeσ3
m=0, α/σ3=0.06, α0=0
0 10 20 30 40
βm0
2/σ3
0
50
100
150
200-β
µ p
slope=apσ3
α0=0, α=0, βm2/σ3=5.0
TRF band-shape
-6 -4 -2 0
ω/eV
0
0.5
1
1.5
I TR
F(ω)
1.0
0.80.6
0.40.2C2(t)=
m01
=6 D
α01=30 ų
m02
=15 Dα02=50 Å
³
ITRF(ω, t) ∝ e−β|α(t)||ω−Ω0|I1
(
2β√
|α(t)|3λ(t)|ω − Ω0|)
,
JCP 2001, 115, 8933
Time-Resolved Correlation Functions
0 0.2 0.4 0.6 0.8 1C2(t)
0
0.2
0.4
0.6
0.8
1
S Ω,2(t
)
0 0.2 0.4 0.6 0.8 1C2(t)
0
1
2
λ(t)
/λ(∞
)
0 0.2 0.4 0.6 0.8 1
α02=10 ų
α02=40 ų
α02=50 ų
α01=30 ų
g (1)
e (2)t = 0
t = ∞
SΩ,2(t) =〈Ω(t)〉2 − 〈Ω(∞)〉2〈Ω(0〉2 − 〈Ω(∞)〉2
C2(t) =〈δΩ(t)δΩ(0)〉2〈δΩ(0)2〉2
SΩ(t) is NOT a good probe of nonlinear dynamics
Coumarin-153
0 0.5 10
0.5
1
0 0.5 10
0.5
1
S0
S1
m01
=7.4 D
m02
=14.9 D
α01=25.8 ų
α02=30.2 ų
m12
=5.8 D ∆m=7.53 D
0.5 1 1.5 2 2.5h∆ν
st, eV
0
2
4
6
βσi2
for ∆α0 > 0emission
absorption
Stokes
Q-model prediction
4 4.4 4.8 5.2 5.6 6 6.4
∆−νst/kK
8
10
12
14
βσ2 /k
K
solvent polarity
absorption
emission
mDk
mAm
m DA
D
Ak
m
Model and Physical Picture
18 20 22 24
−ν/kK
0
5
10
Inte
nsity
polarizable
em. abs.
two-spherehybrid
Equilibrium configuration
Nonequilibrium configurationδ
δn
1 − n
D
A−m
12. Rp
∆E00
∆qn
∆q0n
∆qn = ∆q0n− δn∆γn
2κn
change in population
Hybrid Model
• Coupling between the D and A states is explicitely considered
• Coupling to all other states is accounted through the dipolar
polarizability
α0 = α0 − 2|m12|
2
∆EDA
Solute-solvent coupling:
−m01 ·R− 12 α01R
2
−m12 ·R −m02 ·R− 12 α02R
2
−m12 ·R
non-Condon coupling
Electron-phonon coupling:
∑
n γ1nqnn1 0
0∑
n γ2nqn
n2
electronic population
Calculation Procedure
20 30
ν/kK
0
0.1
0.2
Inte
nsity
solid - 2MB
dashed - gas phase
abs.em.
FCWD(ν) =∫∞
−∞dx⟨
(δn(R)−1δ(ν − x − ∆E[R])FCWDref(νref + x/δn(R))⟩
10 15 20 25 30
ν/kK
0
1
2
Inte
nsity
abs.em.
acetonitrile
Coumarin-153 band-shapes
12 16 20 24 28 32−ν/kK
0
0.1
0.2
12 16 20 24 28 32−ν/κΚ
0
0.1
0.2
acet
acn
em. abs.
em. abs.
experiment
theory
4.5 5 5.5 6
∆−νst/kK
-2
-1
0
1
2
rela
tive
wid
th, k
K
em
abs
4.6 4.8 5 5.2 5.4 5.6
∆−νst(calc)/kK
0
1
2
3
4
5
∆− ν st s,v
/kK
solvent-induced
vibrational
Spectral intensity and the Franck-Condon factor
Lax, Kubo-Toyozawa, Davydov, 50’s. Spectral intensity:
Iabs/em(ν) ∝ |m12|2FCWD(diagonal matrix elements),
m12 is the transition dipole arising from the interaction with the external
electric field of the radiation
−m12 ·E0(t)
In a polar medium,
−m12 ·R
R is the solvent local field.
Iabs/em(ν) ∝ |m12|2FCWD(m12)
Hole Transfer in DNA
kET ∝ V (R)2 exp[−Ea(R)/kT ]
Experiment:
kET ∝ exp[−βDARDA]
βDA = βV + βλ, βλ =1
4kT
∂λ
∂R
βV ' 0.7−1.7 A−1, βλ = 1.0 A−1
Hole Donor
Hole Acceptor
H 2O
Mg2+
βDA(Exp) ' 0.9A−1
Energy Gap Law
0.5 1 1.5 2λ
s, eV
0
0.1
0.2
0.3
0.4
0.5
Fac
t , eV Marcus-Hush
Q-Model
α ' 1.2 − 1.3
-4 -3 -2 -1 0∆F
0, eV
15
20
25
30
ln(k
ET)
L
G C
LT A
G CT A
CR CS
F act = |α|(
√
∆F0 − λ1α2/(1 + α) −√
|α|λ1
)2
βλ(Marcus − Hush) = 1.0 A−1, βλ(Q − Model) = 0.26 A−1
JPCB 107, 2003, 14509-14520.
Conclusions
-100 0 100βX
0
40
80
β(F i(X
)-F
01)
βλ1=40α1 =−42 1
X0
• 3-parameter model: λ1, λ2, ∆F0
-6 -4 -2 0
ω/eV
0
0.5
1
1.5
I TR
F(ω)
1.0
0.80.6
0.40.2C2(t)=
m01
=6 D
α01=30 ų
m02
=15 Dα02=50 Å
³
• Time-resolved band-shapes: λ(t) 6= Const.
4 4.4 4.8 5.2 5.6 6 6.4
∆−νst/kK
8
10
12
14
βσ2 /k
K
solvent polarity
absorption
emission
• D-A coupling + polarizability = band-shapes
of intense transitions.
-4 -3 -2 -1 0∆F
0, eV
15
20
25
30
ln(k
ET)
L
G C
LT A
G CT A
CR CS
• Energy gap law with λ1 6= λ2.