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:
