A. Nitzan, Tel Aviv University

IAS HU

Tutorial: Electron transferJerusalem, July 2012

1.1. Relaxation, reactions and timescalesRelaxation, reactions and timescales

2.2. Electron transfer in condensed Electron transfer in condensed molecular systemsmolecular systems

3.3. Fundamentals of molecular conductionFundamentals of molecular conduction

IAS Workshop 2012

(1) Relaxation and reactions in condensed molecular systems•Timescales•Relaxation•Solvation•Activated rate processes•Low, high and intermediate friction regimes•Transition state theory•Diffusion controlled reactions

The importance of The importance of timescalestimescales

Molecular processes in Molecular processes in condensed phases and condensed phases and

interfacesinterfaces•Diffusion

•Relaxation

•Solvation

•Nuclear rerrangement

•Charge transfer (electron and xxxxxxxxxxxxxxxxproton)

•Solvent: an active spectator – energy, friction, solvation

Molecular timescales

Electronic 10-16-10-15s

Vibraional period 10-14s

Vibrational xxxxrelaxation 1-10-12s

Diffusion D~10-5cm2/s 10nm 10-7 - 10-8 s

Chemical reactions xxxxxxxxx1012-10-12s

Rotational period 10-12s

Collision times 10-12s

Frequency dependent Frequency dependent frictionfriction

~ cˆ ˆ onstant( ) (0)if

T

t

i tf ik d tte F F

ˆ ˆ~ ( ) (0)ifi t

f i Tk dte F t F

1

DWIDE BAND APPROXIMATION

MARKOVIAN LIMIT

1 /ˆ ˆ~ ( ) (0) ~if Di t

f i Tk dte F t F e

Molecular vibrational Molecular vibrational relaxationrelaxation

Relaxation in the X2Σ+ (ground electronic state) and A2Π (excite electronic state) vibrational manifolds of the CN radical in Ne host matrix at T=4K, following excitation into the third vibrational level of the Π state. (From V.E. Bondybey and A. Nitzan, Phys. Rev. Lett. 38, 889 (1977))

Dielectric solvationDielectric solvation

q = + e q = + eq = 0

a b c

C153 / Formamide (295 K)

Wavelength / nm

450 500 550 600

Rel

ativ

e E

mis

sion

Int

ensi

ty

ON O

CF3

Emission spectra of Coumarin 153 in formamide at different times. The times shown here are (in order of increasing peak-wavelength) 0, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, and 50 ps (Horng et al, J.Phys.Chem. 99, 17311 (1995))

2 11 1 2eV (for a charge)

2 s

q

a

Born solvation energy

““real” solvationreal” solvationThe experimental solvation function for water using sodium salt of coumarin-343 as a probe. The line marked ‘expt’ is the experimental solvation function S(t) obtained from the shift in the fluorescence spectrum. The other lines are obtained from simulations [the line marked ‘Δq’ –simulation in water. The line marked S0 –in a neutral atomic solute with Lennard Jones parameters of the oxygen atom]. (From R. Jimenez et al, Nature 369, 471 (1994)).

“Newton”

dielectric

Electron solvationElectron solvationThe first observation of hydration dynamics of electron. Absorption profiles of the electron during its hydration are shown at 0, 0.08, 0.2, 0.4, 0.7, 1 and 2 ps. The absorption changes its character in a way that suggests that two species are involved, the one that absorbs in the infrared is generated immediately and converted in time to the fully solvated electron. (From: A. Migus, Y. Gauduel, J.L. Martin and A. Antonetti, Phys. Rev Letters 58, 1559 (1987)

Quantum solvation

(1) Increase in the kinetic energy (localization) – seems NOT to affect dynamics

(2) Non-adiabatic solvation (several electronic states involved)

C153 / Formamide (295 K)

Wavelength / nm

450 500 550 600

Rel

ativ

e E

mis

sion

Int

ensi

ty

ON O

CF3

Activated rate processesActivated rate processes

E B

r e ac t i o nc o o r di nate

KRAMERS THEORY:

Low friction limit

High friction limit

Transition State theory

0 /

2B B

TSTE k Tk e

0 /

2B BB B

TSTE k Tk e k

/0

B BE k TB

B

k J ek T

(action)

4k DR0

B

Effect of solvent frictionEffect of solvent friction

A compilation of gas and liquid phase data showing the turnover of the photoisomerization rate of trans stilbene as a function of the “friction” expressed as the inverse self diffusion coefficient of the solvent (From G.R. Fleming and P.G. Wolynes, Physics Today, 1990). The solid line is a theoretical fit based on J. Schroeder and J. Troe, Ann. Rev. Phys. Chem. 38, 163 (1987)).

TST

The physics of transition The physics of transition state ratesstate rates

0

2BEe

0

( ,TST B f BP xk d P x

v v v v)

212

212

0 1

2

m

m

d e

md e

v

v

vv

v

20exp

( )2exp ( )

B

B

B EB E

E mP x e

dx V x

Assume:

(1) Equilibrium in the well

(2) Every trajectory on the barrier that goes out makes it

E B

0

B

r e ac t i o nc o o r di nate

The (classical) transition The (classical) transition state rate is an upper state rate is an upper

boundbound

E B

r e ac t i o nc o o r di nate

•Assumed equilibrium in the well – in reality population will be depleted near the barrier

•Assumed transmission coefficient unity above barrier top – in reality it may be less

R *

a b

diabatic

R *

1

1

2

Adiabatic

*

0

( , )k dR R P R R

Quantum considerations

1 in the classical case( )b aP R

IAS Tutorial 2012

(1) Relaxation and reactions in condensed molecular systems•Timescales•Relaxation•Solvation•Activated rate processes•Low, high and intermediate friction regimes•Transition state theory•Diffusion controlled reactions

IAS Tutorial 2012

(2) Electron transfer processes•Simple models•Marcus theory•The reorganization energy•Adiabatic and non-adiabatic limits•Solvent controlled reactions•Bridge assisted electron transfer•Coherent and incoherent transfer•Electrode processes

Theory of Electron TransferTheory of Electron Transfer

Rate – Transition state theoryRate – Transition state theory

0

( ,TST B abP xk d P

v v v v)

BoltzmannBoltzmannActivation Activation energyenergy

Transition Transition probabilityprobability

Electron transfer in polar Electron transfer in polar mediamedia

•Electrons are much faster than nuclei

Electronic transitions take place in fixed nuclear configurations

Electronic energy needs to be conserved during the change in electronic charge density

a

q = 0

b

q = + e

c

q = + e

Electronic transition

Nuclear relaxation

q = 1q = 0 q = 0q = 1

Electron transfer

Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations

Nuclear motion

Nuclear motion

q= 0q = 1q = 1q = 0

Electron transferElectron transfer

E a

E b

E

e ne r g y

ab

X a X tr X b

Solvent polarization coordinate

EA

Transition state theoryTransition state theory of of electron transferelectron transfer

Adiabatic and non-adiabatic ET processesE

R

E a(R )

E b(R )

E 1(R )

E 2(R )

R *

tt= 0

V ab

Landau-Zener problem

*

0

( , ) ( )b ak dRR P R R P R

2,

*

2 | |( ) 1 exp a b

b a

R R

VP R

R F

*

2,| |

2Aa b E

NAR R

VKk e

F

(For harmonic diabatic surfaces (1/2)KR2)

Electron transfer – Electron transfer – Marcus theoryMarcus theory

(0) (0) (1) (1)B BA Aq q q q (0) (0) (1) (1)

B BA Aq q q q

D 4

E D 4 P

eP P Pn

1

4e

eP E

4s e

nP E

They have the following characteristics:(1) Pn fluctuates because of thermal motion of solvent nuclei.(2) Pe , as a fast variable, satisfies the equilibrium relationship (3) D = constant (depends on only)Note that the relations E = D-4P; P=Pn + Pe are always satisfied per definition, however D sE. (the latter equality holds only at equilibrium).

We are interested in changes in solvent configuration that take place at constant solute charge distribution

D Es

q = 1q = 0 q = 0q = 1

q= 0q = 1q = 1q = 0

Electron transfer – Electron transfer – Marcus theoryMarcus theory

0 (0) (0)BAq q

(0) (0) (1) (1)B BA Aq q q q (0) (0) (1) (1)

B BA Aq q q q

Free energy associated with a nonequilibrium fluctuation of Pn

“reaction coordinate” that characterizes the nuclear polarization

q = 1q = 0 q = 0q = 1

q= 0q = 1q = 1q = 0

1 (1) (1)A Bq q

The Marcus parabolasThe Marcus parabolas

0 1 0( ) Use as a reaction coordinate. It defines the state of the medium that will be in equilibrium with the charge distribution . Marcus calculated the free energy (as function of ) of the solvent when it reaches this state in the systems =0 and =1.

20 0( )W E 21 1( ) 1W E

21 1 1 1 1

2 2e s A B AB

qR R R

Electron transfer: Electron transfer: Activation energyActivation energy

2[( ) ]

4b a

A

E EE

21 1 1 1 1

2 2e s A B AB

qR R R

E aE A

E b

E

e ne r g y

ab

a= 0 trb= 1

2( )a aW E

2( ) 1b bW E

Reorganization energy

Activation energy

Electron transfer: Effect of Electron transfer: Effect of Driving (=energy gap)Driving (=energy gap)

Experimental confirmation of the inverted regime

Marcus papers 1955-6

Marcus Nobel Prize: 1992

Miller et al, JACS(1984)

Electron transfer – the Electron transfer – the couplingcoupling

• From Quantum Chemical Calculations

•The Mulliken-Hush formula max 12DA

DA

VeR

• Bridge mediated electron transfer

2 4~

ab

B

E

k Tet abk V e

Bridge assisted electron Bridge assisted electron transfertransfer

D A

B 1 B 2 B 3

D A

12

3V D 1

V 1 2 V 2 3

V 3 A

1

1 1

1

, 1 , 11

ˆ

1 1

1 1

N

D j Aj

D D AN NA

N

j j j jj

H E D D E j j E A A

V D V D V A N V N A

V j j V j j

, 1 /,j B j j B D AE E V E E

EB

VDB

D A

BVAD E

D A

Veff DB ABeff

V VV

E

VDB

D A

B1

VAD

D A

E

Veff

1eff DB N ABV V G V

B2 BN…V12

12 23 1,1

... N NN N

V V VG

E

1

1

1exp (1 / 2) '

NB

N NB

VG N

E V

' 2 ln / BE V D A

12

3V D 1

V 1 2 V 2 3

V 3 A

Green’s Function

1ˆG E E H

Marcus expresions for non-Marcus expresions for non-adiabatic ET ratesadiabatic ET rates

2

2 (1

2)1 ( )

|

)2

| ( )

(

2

BD

DA

D

D A AD

N ANA D

V

V

E

GV E

k

E

F

F

2 / 4

( )4

BE k T

B

eE

k T

F

Bridge Green’s Function

Donor-to-Bridge/ Acceptor-to-bridge

Franck-Condon-weighted DOS

Reorganization energy

Bridge mediated ET rateBridge mediated ET rate

~ ( , )exp( ' )ET AD DAk E T RF

’ (Å-1)=

0.2-0.6 for highly conjugated chains

0.9-1.2 for saturated hydrocarbons

~ 2 for vacuum

Bridge mediated ET rateBridge mediated ET rate(J. M. Warman et al, Adv. Chem. Phys. Vol 106, 1999).

Incoherent hoppingIncoherent hopping

........

0 = D

1 2 N

N + 1 = A

k 2 1

k 1 0 = k 0 1 e x p (-E 1 0 ) k N ,N + 1 = k N + 1 ,N e x p (-E 1 0 )

0 1,0 0 0,1 1

1 0,1 2,1 1 1,0 0 1,2 2

1, 1, , 1 1 , 1 1

1 , 1 1 1,

( )

( )N N N N N N N N N N N N

N N N N N N N

P k P k P

P k k P k P k P

P k k P k P k P

P k P k P

constant STEADY STATE SOLUTION

ET rate from steady state ET rate from steady state hoppinghopping

........

0 = D

1 2 N

N + 1 = A

k

k 1 0 = k 0 1 e x p (-E 1 0 ) k N ,N + 1 = k N + 1 ,N e x p (-E 1 0 )

k k

/

1,0

1

1

B BE k T

D A N

N A D

kek k

k kN

k k

Dependence on Dependence on temperaturetemperature

The integrated elastic (dotted line) and activated (dashed line) components of the transmission, and the total transmission probability (full line) displayed as function of inverse temperature. Parameters are as in Fig. 3 .

The photosythetic reaction The photosythetic reaction centercenter

Michel - Beyerle et al

Dependence on bridge Dependence on bridge lengthlength

Ne

11 1up diffk k N

DNA (Giese et al 2001)DNA (Giese et al 2001)

(2) Electron transfer processes•Simple models•Marcus theory•The reorganization energy•Adiabatic and non-adiabatic limits•Solvent controlled reactions•Bridge assisted electron transfer•Coherent and incoherent transfer•Electrode processes

AN, Oxford University Press, 2006

IAS Tutorial 2012

IAS Tutorial 2012(3) Molecular conduction•Simple models for molecular conductions•Factors affecting electron transfer at interfaces•The Landauer formula•Molecular conduction by the Landauer formula•Relationship to electron-transfer rates.•Structure-function effects in molecular conduction•How does the potential drop on a molecule and why this is important•Probing molecules in STM junctions•Electron transfer by hopping

Molecular conductionMolecular conduction

m o lecule

Molecular Rectifiers

Arieh Aviram and Mark A. RatnerIBM Thomas J. Watson Research Center, Yorktown Heights, New

York 10598, USADepartment of Chemistry, New York New York University, New

York 10003, USA

Received 10 June 1974Abstract

The construction of a very simple electronic device, a rectifier, based on the use of a single organic molecule is discussed. The molecular rectifier consists of a donor pi system and an acceptor pi system, separated by a sigma-bonded (methylene) tunnelling bridge. The response of such a molecule to an applied field is calculated, and rectifier properties indeed appear.

Xe on Ni(110)

m ole c ule

•Fabrication

•Characterization

•Stability

•Funcionality

•Control

•Fabrication

•Stability

•Characterization

•Funcionality

•Control

H O M O

L U M O

EF

Workfunction

System open to electrons and energy

THE MOLECULE

Nonequilibrium

Electron-vibration coupling

Heat generation

Relaxation

Strong electric field

Landauer formulaLandauer formula2

( 0) ( ) ; Fermi energye

g E

T

( ) ( ) ( )L R

eI dE f E f E E

T ( )

dIg

d

1 1

2 21 1

( ) ( )( )

( ) / 2

L RE EE

E E E

T

(maximum=1)

2

112.9

eg K

Maximum conductance per channel

For a single “channel”:

( ))2

( ) (L Rf Ee

I d fE EE

T

fL(E) – fR(E) T(E)

e

fL(E) – fR(E)

T(E)e

I

Weber et al, Chem. Phys. 2002

g

The N-level bridge (n.n. The N-level bridge (n.n. interactions)interactions)

0

{ r }{ l}

RL

1 . . . . N + 1

2

( )e

g E

T

( ) ( )20, 1 0 1( ) | ( ) | ( ) ( )L R

N NE G E E E T

( ) ( ) ( )L R

eI dE f E f E E

T

01 12 , 10, 1

1 10

1 1 1 1ˆ ( ) ...B N NN

N N

G E V V VE E E E E EE E

0 0

1

1

2 LE E i

1 1,

1

1

2N N RE E i

G1N(E)

1 1

2 21 1

( ) ( )( )

( ) / 2

L RE EE

E E E

T

Electron Transfer vs Electron Transfer vs ConductionConduction

2

01 ,(11

2

2)

2| |

2

(

(( )

)

)

AD

NBN D

A DA

AN

D

DV V

E

G E

k

E

V

F

F

01 , 1

( ) ( )0

( ) ( )

1

0 1

( ) ( )0

2

2

( )

1

1

2

1

0,

2

2

| ( ( ) ( )

( )1 1

)

)

2

(

2

( )

|

N N

L RD

L RN

L RNN

A

B

N

N

eg

e V V

E E i E

E E

E

G

GE

EE

E

i

........

0 = D

1 2 N

N + 1 = A

E

2 / 4

( )4

BE k T

B

eE

k T

F

A relation between g and A relation between g and kk

2

2 ( ) ( )

8D AL R

D A

eg k

F

conduction Electron transfer rate

MarcusDecay into electrodes

Electron charge

A relation between g and A relation between g and kk

2

2 ( ) ( )

8D AL R

D A

eg k

F

1

4 exp / 4B Bk T k T

F

eV( ) ( ) 0.5L RD A eV

2 13 1

17 1 1

~ / 10 ( )

10 ( )

D A

D A

g e k s

k s

Alkane Bridge§ X(CH2)n-2

low bias limit

I / V in nano-pore junctions

Reed et al(monothiolates)

STM / break junctions Tao et al (dithiolates)

Scaled k0: ‡

5 x 10-19 α k0/DOS*

Nitzan M(DBA)M

model ( D and A chemisorbed to M)

n=8 5.0 E-11 1.9 E-8 4.1 α E-8

n=10 5.7 E-12 1.6 E-9 6.8 α E-9

n=12 6.5 E-13 1.3 E-10 4.6 α E-10Conclusions: • conductance data of Tao et al (g) and rate constant data (k0) correspond to within ~ 1-2 orders of magnitude

• results from the 2 sets of conductance measurements differ by > 2 orders of magnitude

Conductance (g (Ω-1)) vs Kinetics ( k0 (s-1) ) for alkane spacers [Marshal Newton]

Temperature and chain Temperature and chain length dependencelength dependence

Giese et al, 2002

Michel-Beyerle et al

Xue and Ratner 2003

M. Poot et al (Van der Zant), Nanolet 2006

Barrier dynamics effects on Barrier dynamics effects on electron transmission electron transmission

through molecular wiresthrough molecular wires

•HEAT CONDUCTION -- RECTIFICATION

•INELASTIC TUNNELING SPECTROSCOPY

•STRONG e-ph COUPLING: (a) resonance inelastic tunneling spectroscopy (b) multistability and hysteresis

•NOISE

•RELEVANT TIMESCALES

•INELASTIC CONTRIBUTIONS TO CURRENT

•DEPHASING AND ACTIVATION

•HEATING

AN, Oxford University Press, 2006

SUMMARY(1) Relaxation and reactions in condensed molecular systems•Kinetic models•Transition state theory•Kramers theory and its extensions•Low, high and intermediate friction regimes•Diffusion controlled reactions

(2) Electron transfer processes•Simple models•Marcus theory•The reorganization energy•Adiabatic and non-adiabatic limits•Solvent controlled reactions•Bridge assisted electron transfer•Coherent and incoherent transfer•Electrode processes

(3) Molecular conduction•Simple models for molecular conductions•Factors affecting electron transfer at interfaces•The Landauer formula•Molecular conduction by the Landauer formula•Relationship to electron-transfer rates.•Structure-function effects in molecular conduction•Electronic conduction by hopping•Inelastic tunneling spectroscopyChapter 13-15Chapter 16Chapter 17

THANK YOUA. Nitzan, Tel Aviv University

INTRODUCTION TO ELECTRON TRANSFER AND MOLRCULAR CONDUCTION

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