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.