Lecture 1 Theoretical models for transport, transfer and relaxation in molecular systems A. Nitzan,...

Lecture 1Lecture 1Theoretical models for transport, transfer Theoretical models for transport, transfer

andandrelaxation in molecular systemsrelaxation in molecular systems

A. Nitzan, Tel Aviv University

SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS

INTRODUCTIONINTRODUCTION

Chemical dynamics in condensed phases

Molecular relaxation processes

•Quantum dynamics•Time correlation functions•Quantum and classical dissipation•Density matrix formalism•Vibrational relaxation•Electronic relaxation (radiationaless transitions)

•Solvation•Energy transfer•Applications in spectroscopy

Condensed phases Molecular reactions

Quantum dynamicsTime correlation functionsStochastic processesStochastic differential equationsUnimolecular reactions: Barrier crossing processesTransition state theoryDiffusion controlled reactionsApplications in biology

Electron transfer and molecular conduction

Quantum dynamicsTunneling and curve crossing processesBarrier crossing processes and transition state theoryVibrational relaxation and Dielectric solvationMarcus theory of electron transferBridge assisted electron transferCoherent and incoherent transferElectrode reactionsMolecular conductionApplications in molecular electronics

electron transport in molecular electron transport in molecular systemssystems

Reviews:

Annu. Rev. Phys. Chem. 52, 681– 750 (2001) Science, 300, 1384-1389 (2003); J. Phys.: Condens. Matter 19, 103201 (2007) – Inelastic effects

Phys. Chem. Chem. Phys., 14, 9421 - 9438 (2012) – optical interactions

Molecular PlasmonicsSolar cells, OLEDs

Chemical processesChemical processes

Gas phase Gas phase reactionsreactions

Follow individual Follow individual collisionscollisions

States: InitialStates: InitialFinal Final Energy flow between Energy flow between

degrees of freedomdegrees of freedom Mode selectivityMode selectivity Yields of different Yields of different

channelschannels

Reactions in Reactions in solutionsolution

Effect of solvent on Effect of solvent on mechanismmechanism

Effect of solvent on Effect of solvent on ratesrates

Dependence on Dependence on solvation, solvation, relaxation, diffusion relaxation, diffusion and heat transport.and heat transport.

I2 I+I

A.L. Harris, J.K. Brown and C.B. Harris, Ann. Rev. Phys. Chem. 39, 341(1988)

molecular absorption at ~ 500nm is first bleached (evidence of depletion of ground state molecules) but recovers after 100-200ps. Also some transient state which absorbs at ~ 350nm seems to be formed. Its lifetime strongly depends on the solvent (60ps in alkane solvents, 2700ps (=2.7 ns) in CCl4). Transient IR absorption is also observed and can be assigned to two

intermediate species .

The hamburger-dog dilemma as a lesson in the importance of timescales

1 0 -1 5 1 0 -1 4 1 0 -1 3 1 0 -1 2 1 0 -1 1 1 0 -1 0 1 0 -9 1 0 -8

T I M E (s e c o n d )

v ib ra tio n a l m o tio n

e le c tro nicde pha s ing v ibra tio na l de pha s ing

v ib ra tio n a l re la x a tio n (p o lya to m ic s )e le c tro nic re la xa tio n

c o llis io n tim ein liq u id s

so lv e nt re la xa tio n

m o le c ula r ro ta tio n

p r o to n tr a n sfe rp r o te in in te r n a l m o tio n

e n e rg y tra n s fe r inp h o to s yn th e s is

T o rs io n a ld yn a m ic s o f

D N A

e le c tro n tra ns fe rin pho to s ynthe s is

pho to io niza tio npho to disso c ia tio n

p h o to c h e m ic a l iso m e r iza tio n

TIMESCALES

Typical molecular timescales in chemistry and biology (adapted from G.R. Fleming and P. G. Wolynes, Physics today, May 1990, p. 36) .

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

Diffusion D~10-5cm2/s

Electronic 10-16-10-15s

Vibraional 10-14s

Vibrational xxxxrelaxation 1-10-12s

Chemical reactions xxxxxxxxx1012-10-12s

Rotational 10-12s

Collision times 10-12s

VIBRATIONAL VIBRATIONAL RELAXATIONRELAXATION

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

Golden Rule 22k V

Molecular vibrational Molecular vibrational relaxationrelaxation

1large (~1ps ) and

weakly dependent n

oVRk

~ D

c

VRk e

“ENERGY GAP LAW”

kVR

D

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))

Molecular vibrational Molecular vibrational relaxationrelaxation

The relaxation of different vibrational levels of the ground electronic state of 16O2 in a solid Ar matrix. Analysis of

these results indicates that the relaxation of the < 9 levels is dominated by radiative decay and possible transfer to impurities. The relaxation of the upper levels probably takes place by the multiphonon mechanism. (From A. Salloum, H. Dubust, Chem. Phys.189, 179 (1994)).

DIELECTRIC DIELECTRIC SOLVATIONSOLVATION

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

Continuum dielectric theory of Continuum dielectric theory of solvationsolvation

D 4

(r, ) ( ) (r, )t

D t dt t t E t

D E

( ) ( ) 4 ( )

( ) ( ) ( )

1

4

D E P

P E

D(r, ) r ' (r r ', )E(r ', )εt

t d dt t t t

1 2

( )1

s ee

Di

How does solvent respond to a sudden change in the molecular charge distribution?

Electric displacement

Electric field

Dielectric function

Dielectric susceptibility

polarizationDebye dielectric relaxation model

Electronic response

Total (static) response

Debye relaxation time

(Poisson equation)

Continuum dielectric theory of Continuum dielectric theory of solvationsolvation

0 0( )

0

tE t

E t

1; 0s

D D

dDD E t

dt

1( ) ( 4) ;e s

D

dE

dDD D E

t

/ /( ) (1 )D Dt ts eD t e e E

0 0( )

0

tD t

D t

1; 0s

e D s

dE E D t

dt

/1 1 1( ) Lt

s e s

E t D De

eL D

s

WATER:

D=10 ps L=250 fs

““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

Electron tunneling Electron tunneling through waterthrough water

E F

W o rkfu n ct io n( in wa te r)

W A T E R

12

3

Polaronic state (solvated electron)

Transient resonance through “structural defects”

Electron tunneling Electron tunneling through waterthrough water

Time (ms)

STM current in pure waterSTM current in pure waterS.Boussaad et. al. JCP (2003)S.Boussaad et. al. JCP (2003)

CHEMICAL CHEMICAL REACTIONS IN REACTIONS IN CONDENSED CONDENSED

PHASESPHASES

Chemical reactions in Chemical reactions in condensed phasescondensed phases

Bimolecular

Unimolecular

diffusion

4k DR

Diffusion controlled

rates

Bk TD

mR

2

1

k1 2 k2 1

k2

excitation

reaction

21 2

12 2

k Mkk

k M k

k

M

Thermal interactions

Unimolecular reactions (Lindemann)

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)

0

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

What we covered so far

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

Electron transfer

Electron transfer in polar Electron transfer in polar mediamedia

•Electron 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

c

q = + e

b

q = + e

a

q = 0

Electronic transition

Nuclear relaxation (solvation)

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

Electron transfer

ELECTRONIC ENERGY CONSERVED

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 aE A

E b

E

e ne r g y

ab

X a X tr X b

Solvent polarization coordinate

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

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

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

Alternatively – solvent control

Solvent controlled electron Solvent controlled electron transfertransfer

Correlation between the fluorescence lifetime and the longitudinal dielectric relaxation time, of 6-N-(4-methylphenylamino-2-naphthalene-sulfon-N,N-dimethylamide) (TNSDMA) and 4-N,N-dimethylaminobenzonitrile (DMAB) in linear alcohol solvents. The fluorescence signal is used to monitor an electron transfer process that precedes it. The line is drawn with a slope of 1. (From E. M. Kosower and D. Huppert, Ann. Rev. Phys. Chem. 37, 127 (1986))

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

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)

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

SUMMARY

IRREVERSIBILITYIRREVERSIBILITY

What is the source of irreversibility in the processes discussed?

• Vibrational relaxation• Activated barrier crossing• Dielectric solvation• Electron transfer

V

0

V0l

l

Starting from state 0 at t=0:

P0 = exp(-t)

= 2|V0l|2L (Golden Rule)

2cos 2 /V

Steady state evaluation Steady state evaluation of ratesof rates

Rate of water flow depends linearly on water height in the cylinder

Two ways to get the rate of water flowing out:

(1) Measure h(t) and get the rate coefficient from k=(1/h)dh/dt

(1) Keep h constant and measure the steady state outwards water flux J. Get the rate from k=J/h

= Steady state rate

h

Steady state quantum Steady state quantum mechanicsmechanics

{ }l l0

V0l

Starting from state 0 at t=0:

P0 = exp(-t)

= 2|V0l|2L (Golden Rule)

Steady state derivation:

0 0 0 sl ll

dC iE C i V C

dt

0( ) ( ) 0 ( )ll

t C t C t l 0( / )

0 0i E tC c e

0 0 ; alll l l ld

C iE C iV C ldt

l

0( / )0 0

i E tC c e

0 0 ; alll l l ld

C iE C iV C ldt

(1 / 2) lC

0( / ) 0

0

( ) ;/ 2

i E t lsl l l

l

V cC t c e c

E E i

2 2 20 0 2 2

0

2 200 0 0

//

/ 2

2

l ll l l

l ll

J C C VE E

C V E E

0

020

2 20 0 0

2 2/

l

l l l LE El

V E E VJ

Ck

pumping

damping{ }l

0

V0l

l

22* *

0 0 0 0l

l l l l ld c

iV c c iV c c cdt

Resonance scatteringResonance scattering

{ }l1

V1r

r l

V1l

0

0H H V

0 0 10

0 0 1 1 l rl r

H E E E l l E r r

0,1 1,0 ,1 1, ,1 1,0 1 1 0 1 1 1 1l l r r

l r

V V V V l V l V r V r

Resonance scatteringResonance scattering

0( / )0 0

i E tC c E

( / 2) rC

( / 2) lC

1 1 1 1,0 0 1, 1,

,1 1

,1 1

l l r rl r

r r r r

l l l l

C iE C iV C i V C i V C

C iE C iV C

C iE C iV C

0( ) exp ( / )j jC t c i E t j = 0, 1, {l}, {r}

For each r and l

0 0 0 0,1 1C iE C iV C

0 0( ) ( / ) exp ( / )j jC t i E c i E t

Resonance scatteringResonance scattering

0

0 1 1 1,0 0 1, 1,

0 ,1 1

0 ,1 1

( / )0 0

( /

0

)

/

0

0

2

( 2)

l l r rl r

r r r

l

i E t

r

l ll

i E E c iV c i V c i

C c E

c

c

V c

i E E c iV c

i E E c iV c

,1 1

0 / 2r

rr

V cc

E E i

,1 1

0 / 2l

ll

V cc

E E i

For each r and l

,1 1

0 / 2r

rr

V cc

E E i

1, 1 0 1( )r r RrV c iB E c

1 1

21

2

1

21

1

11

1

( ) (1 / 2) ( )

( ) 2 | | ( )

| | (

| |( )

)( )

r

R R

R r R r

rR

E E

r R rR

r

rr

VB E

E EE i E

E V E

V EE

i

PP dEE E

1 0 1 1,0 0 1, 1,0 l l r rl ri E E c iV c i V c i V c

1 1

wide ban

( ) (

d approximatio

)

n

1/ 2R RB E i

21

1| |

( ) rR

rr

VB E

E E i

SELF

ENERGY

0 ,1 10 ( / 2)r r r ri E E c iV c c

1,0 01

0 1 1 0( / 2) ( )

V cc

E E i E

1 0 1 0 1 0( ) ( ) ( )L RE E E

2 2 2,1 1,0 02 22 2 2 2

0 0 1 1 0

| | | | | || | | |

( ) ( / 2) ( ) / 2

rr r

r

V V cc C

E E E E E

22 1,00 21 0

0 02 20 1 1 0

| | ( )/ | |

( ) / 2

RR r

r

V EJ c c

E E E

21 02 r rV E E

{ }l1

V1r

r l

V1l

0

Resonant tunnelingResonant tunneling

21,0 21

0 02 20 1 1

| || |

/ 2

RR

VJ c

E E

|1 >

|0 >

x

V (x )

RL

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

(a )

(b)

( c)

L

R

{ }l1

V1r

r l

V1l

0

V10

SummarySummary

1

V1r

r lV1l

0

{ }l

1

V1r

r

V01

00

r

V0r

1

V1r

r lV1l0

21,0 21

0 02 20 1 1

| || |

/ 2

RR

VJ c

E E

02

0

20 02

Rt

R R R

c t e

V

Lecture 2Lecture 2electron transfer, energy transfer, electron transfer, energy transfer,

molecular conduction, inelastic molecular conduction, inelastic spectroscopies, heat conduction, spectroscopies, heat conduction,

optical effects…optical effects…

A. Nitzan, Tel Aviv University

TOMORROW: