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Theory vs molecular modelling of charge transfer reactions: some problems and challenges

Renat R. Nazmutdinov
Kazan National Research State Technological University Dubna, August., 28 2012


Outline
1. Motivation
2. Medium (solvent) coordinates

3. Quantum effect of solvent on activation barrier 4. Orbital overlap (electronic transmission coefficient) 5. Some problems ahead
tst


Transition state theory (TST)

saddle point

Three-dimensional reaction energy surface (solvent and intramolecular coordinates)
marc


A simple way to define the solvent coordinate
(Marcus theory)

is the solvent reorganization energy

Free energy (U)

(Ohmine, 1992)




i f

Ui (q) = q

2

Uf (q) = (q -1) + I
2

non-equilibrium solvent coordinate (q)

( +I) Ea = 4 md
2


Computer simulations
U
free energy

UQ) =-kT ln P(Q) ( B
P(Q)

reactant

f

Q product Umbrella sampling

i Coulomb part of the solvation energy (Q)

kram


Stochastic theory

Reaction rate depends on dynamical solvent properties as well (friction, viscosity)

Hendrik A. Kramers /pioneered a stochastic appoach in chemical kinetics/

Leonid D. Zusman In terms of stochastic theory an overcoming /extended Kramers theory of the activation barrier more resembles to electron transfer reactions/

"climbing" (diffusion)

corr


Solvent correlation function

e
E
F

K()= E A (0), E A ()
EA

Q( ) K ( ) = 2k BT s Q(0)


i Q( ) = 2

-



1 1 d - exp ( -i ) ( )
dielectric spectrum
spectra

S. Mukamel et al.


Examples of dielectric spectra sucrose solutions:

C D ( w) = + + 1 + (iw C ) 1 + iw
water-EG mixtures:

D

3 1 2 ( ) = + + + 1 + i 2 1 1 + i 2 2 1 + i 2 3



series


N solvent modes
Solvent reorganization energy

(exact expansion )
correlation times

K ( ) = 2k BT
Solvent correlation function


i =1

N

i exp(- / )

* i


i =1

N

i = 1



i is the contribution of i-th mode to the solvent reorganization energy
The solvent reorganization energy is "distributed" among N solvent coordinates.

afes


Reaction free energy surface can be described using N solvent coordinate (q1, ... qN) and (probably) one intramolecular degree of freedom (r):

Ei (q1 , ..., qN ; r ) =
reactant



N

j =1

j

2 jq j

+ U i (r )

E f (q1 , ..., qN ; r ) =
product



N

j =1

j j (q j - 1) + U f (r ) + I

2

Usually N = 2 (e.g., dimethylacetamide), 3 (EG, alcohols etc)
pers


S2O82- reduction at a mercury electrode from water-EG mixtures

S

2- 2O8

+e =

2- SO4

+

- SO4

- reaction is adiabatic

The first ET is rate limiting

- BBET reaction proceeds at large overvoltages, in the vicinity of activationless discharge, i.e. at small activation barriers
8
potential (V) -0.6 -0.7 -0.8 -0.9 -1.0 -1.1

I , A

4

- reaction reveals an anomalous solvent viscosity effect

0 0 20

%, EG

40

60

Exp. data (P.A. Zagrebin et al) l, md


- Pekar factor in the solvent reorganization energy is nearly constant - MD simulations predict even a slight increase of <> Bulk contributions to the solvent reorganization energy as computed form molecular dynamics (O. Ismailova, M. Probst et al)
70 60
-1



red ox

, kcal mol

50 40 30 20 10

<>

0.0

0.2

0.4

0.6

0.8

1.0

xEG

ks


Results of Langevin molecular dynamics simulations

10 8

,

Ks, ps

-1

)в

6 4 2 0

3.1 3.05 3.0 2.9 2.8

(

10

4

0

20

40

xEG

60

80

100

avoid


An attempt to explain: saddle point avoidance
1.0 0.9 0.8



0.7 0.6 0.5

0

20

40

xEG

60

80

100

prot


Non-Gaussian fluctuations
- ferroelectric domains at a protein/water interface D.N. LeBard, D.V. Matyushov, PCCP, 12 (2010) 15335

IL


MD simulation of the Au(111)/[BMIM][BF4] interface
(S.A. Kislenko et al)

U

Non-linear response ?
q quant


Solvent coordinate vs Quantum effects
- decreasing of the activation barrier - tunneling
decreasing rate constant

increasing rate constant

U

U

q

q eq


Effect of solvent quantum modes
* (s* - ) 2 Ea k = exp - exp[- ] exp[- ] = exp - * k BT 4s k BT

s* = s
2 = C
2s = C
tunneling factor

*


0


Im ( )

( )

2

d

C=
d

1



-

1





st





Im ( )

*

( )
2

2

Pekar factor

mdb


Interpolation ch. . . sh w, w* h by polynomial of degree 6 on the interval 0 , 3w* h
F 1.0 0.8 0.6 0.4 0.2

@ @D 8 @D<
*



= 0.5

0

100

200

300

400

500

w THz



ch{ /2} - ch{(1 - 2 ) / 2} F ( , ) = sh{ / 2}

w

old


Dielectric spectra of water (J.A. Saxton, 1953)

= 0.8

new


R. Buchner and co-workers (2005)

S1 S2 (v ) = + + + 1 + i 2 v 1 1 + i 2 v 2
2 S3 S4v4 +2 2 1 + i 2 v 3 v4 - v + i 4 v 2 2 S5v5 S6v6 +2 2 +2 2 + v5 - v + i 5v v6 - v + i 6 v 2 S7 v6 +2 2 v7 - v + i 7 v

= 0.9
IL


R. Buchner and co-workers (2008)

Dielectric spectra of some ionic liquids

= 0.55

kapp


Electronic transmission coefficient

1 - exp(-2 e ) e = 1 - (1 / 2) exp(-2 e )
Landau-Zener factor

Ee 2 e =

2

half of resonance splitting



eff

(s + in ) k BT

effective frequency

Two important limiting cases:

e << 1 e e >> 1 e 1

e

(non-adiabatic) (adiabatic)

Vif


It is reasonable to employ the perturbation theory for large molecular systems

Ee ^ ^ iV f dV - iV i dV i f dV 2
Perturbation (molecular electrostatic potential)

V ( r ) =-


i

Z

i

Ri - r

+


j

j (r
/

/

)

2

d

/

r -r
ChelpG atomic charges

V (r )


i

qi* Ri - r

cyt c4


Orientation of the cyt c4 heme groups which leads to the maximal intramolecular ET rate

heme B (red)

heme A (ox)

e (max) = 0.2

mc


- ( )

sash


Electronic transmission coefficient vs density of electronic states calculated with the help of MC simulations at different values of e
1.0 0.8 0.6
2e=0.999 2e=0.1 2e=0.01 2e=0.001



e

0.4 0.2 0.0 0 10 20 30

( F )k BT 2
* e

e

40 N

50

60

70

density of electronic states

slab, cl


Model calculation of electronic transmission coefficient for interfacial reactions: some challenges. 1. Model of a charged metal surface (cluster, slabs, "jellium", etc)

2. Solvent effect on the wave functions and perturbation 3. Asymptotic behaviour of wave functions
wave


Electronic density of metal slabs vs distance

n( x ) =
4
n(x)

Me

( x)

2

n( x ) = A exp( - x )

2 0 -4

Me ( x) = A exp(- x / 2)
Effective wave function of metal

-2

0
x, a.u.

2

4

6
Fc


Fc + e

+

Fc
1

= 0 exp(- z )

lg

0.1

= 1.3 A
0.01 8 9

-1

z, A

10

11

Au54 cluster
stm


Model STM contrast

Cysteine adsorption on Au(110) elecrode (in situ STM images)

wire


Me(111) vs monoatomic wires Fe(III)/Fe(II)
60

e(Me(111))/ e(mono)

40

Ag Au Cu Pt

20

0 0.6

0.7

x, nm

0.8

0.9

fin