Electron Transfer in Inorganic, Organic, and Biological Systems

4 Theoretical Analysis of Energy-Gap Laws of Electron-Transfer Reactions Downloaded by COLUMBIA UNIV on March 20, 2013 | http://pubs.acs.org Publication Date: May 5, 1991 | doi: 10.1021/ba-1991-0228.ch004

Distribution Effect of Donor-Acceptor Distance Toshiaki Kaldtani , Akira Yoshimori , and Noboru Mataga 1

1

2

Department of Physics, Faculty of Science, Nagoya University, Chikusaku, Nagoya 464-01, Japan Department of Chemistry, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan

1

2

The electron-transfer rate as a function of the free energy gap (energy-gap hw) was formulated by including the solvent nonlinear response effect and averaging over the distribution of donor-acceptor distances. Using the same parameter values, we fit the theoretical energy-gap laws to three independent experimental measurements: the photoinduced charge-separation (CS) rate as measured from fluorescence quenching in the stationary state, the actual photoinduced charge-separation rate as obtained from analysis of the transient effect in the fluorescence decay curve, and the charge-recombination (CR) rate of the geminate radical-ion pair. The different energy-gap laws among those reactions can be reproduced reasonably well by adopting different distributions of the donor-acceptor distance: that of the CS reaction covering those over various distances and more specified ones corresponding to the contact ion-pair (CIP) and solvent-separated ion-pair (SSIP) models for the CR reaction. The nonlinear effect in those homogeneous reactions is small when the CIP model applies and appreciable when the SSIP model applies.

E L E C T R O N - T R A N S F E R (ET) R E A C T I O N S I N P O L A R S O L U T I O N S and

electrode

systems are a m o n g the most f u n d a m e n t a l c h e m i c a l processes. F r o m a t h e oretical p o i n t of v i e w , i f the linear response of the solvent p o l a r i z a t i o n is

0065-2393/91/0228-0045$07.25/0 © 1991 American Chemical Society

In Electron Transfer in Inorganic, Organic, and Biological Systems; Bolton, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1991.

46

E T IN INORGANIC, ORGANIC, A N D BIOLOGICAL SYSTEMS

Downloaded by COLUMBIA UNIV on March 20, 2013 | http://pubs.acs.org Publication Date: May 5, 1991 | doi: 10.1021/ba-1991-0228.ch004

a s s u m e d , the G i b b s energy-gap d e p e n d e n c e of the E T rate (called the e n ergy-gap law) b e c o m e s b e l l - s h a p e d (1-5). T h e r e is a r a p i d rate increase i n the small-energy-gap r e g i o n (normal region), a m a x i m u m at an e n e r g y gap e q u i v a l e n t to the total reorganization energy, a n d a r a p i d rate decrease i n the large-energy-gap r e g i o n (inverted region). T h e l i n e a r response t h e o r y also p r e d i c t s that the transfer coefficient at the standard electrode p o t e n t i a l i n e l e c t r o c h e m i c a l reactions s h o u l d b e V2 (6). O n the o t h e r h a n d , i f d i e l e c t r i c saturation takes place a r o u n d a c h a r g e d reactant o r p r o d u c t , a n o n l i n e a r solvent p o l a r i z a t i o n response is p r e d i c t e d . I n this case, the G i b b s energy c u r v a t u r e a r o u n d the m i n i m u m i n the i n i t i a l state can be different from that i n the final state (7-9). A s a result, the energy-gap l a w of the charge-separation (CS) reaction (A* · * · D —> A " • · · D ) is b r o a d e n e d , i n a d d i t i o n to a flattening a r o u n d the m a x i m u m (7, 10). T h e i n v e r t e d r e g i o n exists so far as the G i b b s energy c u r v e of the solvent is c o r r e c t l y calculated along the reaction coordinate (11-13). T h e e n e r g y gap l a w of the c h a r g e - r e c o m b i n a t i o n (CR) reaction ( A " · · · D —» A · · · D ) is n a r r o w e d , w i t h a shift of the peak to a smaller e n e r g y gap (11-16). T h e transfer coefficient of the i o n i z a t i o n reaction [A ~ * · * M(e") —> A ~ +

+

z

( z

) _ 1

• · · Μ , ζ b e i n g a valence ^ 0] is smaller t h a n V2 a n d that of the n e u t r a l i z a t i o n reaction [ A · · · M ( e " ) —» A * " · · · Μ , ζ ^ 1] is larger than V2, w h e r e A ~ or A is the reactant a n d M the m e t a l electrode (17). T h e s e results i m p l y that the energy-gap l a w a n d the transfer coefficient are systematically m o d ified b y n o n l i n e a r response, d e p e n d i n g o n the type of the E T reaction (7-11, 14-18). z

1

z

z

W h e n a l i n e a r response applies, the energy-gap l a w does not d e p e n d o n the t y p e of the E T reaction. T h e r e f o r e , the significance of the n o n l i n e a r response effect can b e d e t e r m i n e d b y c o m p a r i n g the energy-gap laws b e t w e e n the C S a n d C R reactions. E x p e r i m e n t a l data have d e m o n s t r a t e d that b e l l - s h a p e d energy-gap laws, w h i c h are o b t a i n e d for the C R reaction of the geminate i o n p a i r p r o d u c e d b y fluorescence q u e n c h i n g (18), are q u a l i t a t i v e l y i n agreement w i t h the p r e d i c t i o n of the l i n e a r response theory. F o r the C S reaction, the i n v e r t e d r e g i o n was not o b t a i n e d u p to an energy gap of 2.5 e V (i.e., the E T rate constant was so large as to b e m a s k e d b y the d i f i u s i o n - l i m i t rate constant of the reactants u n t i l the value of the e n e r g y gap was 2.5 eV) (19). P a r t i c i p a t i o n of the excited-state p r o d u c t was e l i m i n a t e d at a n e n e r g y gap of 1.6 e V (20), b u t it may be possible i n the 1 . 6 - 2 . 5 - e V energy-gap r e g i o n . I n the n o r m a l r e g i o n , t h e r e seems to be a considerable difference b e t w e e n the C S a n d C R reactions. T h e n o r m a l r e g i o n of the C S reaction is c o n s i d e r a b l y shifted t o w a r d a s m a l l energy gap, as c o m p a r e d w i t h that of the C R reaction (18). O n the other h a n d , i n e l e c t r o c h e m i c a l reactions, most of the e x p e r i m e n t a l data o n the transfer coefficient for the i o n i z a t i o n r e a c t i o n are s m a l l e r t h a n ¥2 w h e n the standard rate constant is less t h a n 1 c m s . M o s t data for the n e u t r a l i z a t i o n reaction are larger than V2 (17), consistent 1


4.

KAKITANI ET AL.

Energy-Gap

Laws of Electron-Transfer

Reactions

47

w i t h the t h e o r e t i c a l calculations based o n the n o n l i n e a r response theory. A l l of these e x p e r i m e n t a l data appear to indicate that the energy-gap l a w varies b e t w e e n the C S a n d C R reactions, a n d b e t w e e n the i o n i z a t i o n a n d n e u tralization reactions.


O n e i m p o r t a n t factor was not taken into account i n the foregoing t h e oretical treatments. So far, the distance b e t w e e n the d o n o r a n d acceptor was i m p l i c i t l y a s s u m e d to b e a constant. It m a y be possible for the n e u t r a l d o n o r a n d acceptor to d i s t r i b u t e over various distances. H o w e v e r , the i o n p a i r d i s t r i b u t i o n m i g h t be l o c a l i z e d (i.e., the m a n n e r of the distance d i s t r i b u t i o n can b e different b e t w e e n the C S a n d C R reactions). T h e r e f o r e , it is i m p o r t a n t to investigate h o w such d i s t r i b u t i o n s , i f t h e y occur, can affect the o b s e r v e d energy-gap law. T h i s d i s t r i b u t i o n effect o n the rate constant was first investigated b y M a r c u s a n d Siders (21). B r u n s c h w i g et a l . (22) m a d e n u m e r i c a l calculations for the charge shift reaction of transition m e t a l c o m plexes b y taking into account the distance d e p e n d e n c e of the reorganization e n e r g y a n d the t u n n e l i n g m a t r i x e l e m e n t , as w e l l as the d i s t a n c e - d i s t r i b u t i o n function. W e m a d e a d e t a i l e d investigation as to w h e t h e r the different e n e r g y gap laws of the C S a n d C R reactions can b e r e p r o d u c e d , b y e i t h e r the l i n e a r or n o n l i n e a r response t h e o r y b y taking into account most possible effects attributable to the distance d i s t r i b u t i o n b e t w e e n the d o n o r a n d acceptor. I n such analyses, w e u s e d three i n d e p e n d e n t e x p e r i m e n t a l m e a s u r e m e n t s : the p h o t o i n d u c e d C S rate constant as m e a s u r e d f r o m fluorescence q u e n c h i n g i n the stationary state (19), the actual p h o t o i n d u c e d C S rate constant as o b t a i n e d b y analysis of the transient effect i n fluorescence decay curves (23, 24), a n d the C R rate of the geminate r a d i c a l - i o n p a i r (18). T h e results d e m onstrate that the difference i n the o b s e r v e d energy-gap laws arises m o s t l y f r o m the different d i s t r i b u t i o n s of d o n o r - a c c e p t o r distances b e t w e e n the C S a n d C R reactions. T h e role of the n o n l i n e a r response is not as large i n h o m o g e n e o u s reactions as e x p e c t e d (7, 10, 14-16), a l t h o u g h it is still c o n siderable.

Formulation of the ET Rate for a Fixed Distance T h i s section presents a theoretical f o r m u l a t i o n of the E T rate w h e n the distance b e t w e e n the d o n o r a n d acceptor is fixed. Solvent M o d e . F i r s t , the f o r m u l a for the E T rate w i l l b e d e r i v e d b y c o n s i d e r i n g o n l y the solvent m o d e a n d assuming that t h e r m a l e q u i l i b r i u m is always attained for the solvent m o t i o n i n the i n i t i a l state. T h i s a s s u m p t i o n appears to b e v a l i d for such a solvent as acetonitrile, whose l o n g i t u d i n a l relaxation t i m e τ is as s m a l l as 2 X 10 s (25). T h e solvent m o l e c u l e i n solution does not m o v e o n a definite p o t e n t i a l L

1 3

American Chemical Society

Library 1155 16th S,*.!. In Electron Transfer in Inorganic, Organic, and Biological Systems; Bolton, J., et al.; Washington, O.G. Society: 20011 Washington, DC, 1991. Advances in Chemistry; American Chemical

48

E T IN INORGANIC, ORGANIC, A N D BIOLOGICAL SYSTEMS

surface; its p o t e n t i a l is t h e s u m o f m a n y long-range interactions that

fluctuate

r a n d o m l y . T h e r e f o r e , i t is i m p o s s i b l e to calculate its F r a n c k - C o n d o n factor d i r e c t l y as a n o v e r l a p o f v i b r a t i o n a l w a v e functions. Instead, i t is m o r e a p p r o p r i a t e to calculate t h e G i b b s e n e r g y c u r v e along t h e reaction c o o r d i nate, w h i c h c a n b e d e f i n e d u n a m b i g u o u s l y . H a m i l t o n i a n s H (r) a n d H (r) for polar solutions i n t h e i n i t i a l a n d final l

F

states, r e s p e c t i v e l y , are as follows: ff'(r)

= H ( r ) + E {(r)

(1)

H ( r ) = H>(r) + E f(r)

(2)

r

e


F

e

where r/ \

H

r

I*a efe

V

=

' rj

r

i

zfeÇjLi · r ) id

+

r

\

+ Σ

Σ

#/"

i d

+ VM

d

3

+ v c

(3)

c

i > j

Η

Ρ( ) γ

' rJ

/ zM^i

y

=

\

i

+ Σ

+

zjefaj · r )

η/

r& Σ

^

ià

d

"

d

+ V "

+

c

(4)

c

i > j

Mr) = ^ EAr)

r

=

+

(5)

+ eP

(6)

and ο

d-d _

&

' ϊ%

3(μ, - r^fcj

·

r) ff

(7)

w h e r e t h e superscripts r a n d ρ d e n o t e reactant a n d p r o d u c t , r e s p e c t i v e l y ; £ > d> a > r

a

z

z

P

a

n

d Za

pa

r

e

valences o f the acceptor a n d d o n o r o f reactants a n d

p r o d u c t s , r e s p e c t i v e l y ; e is a u n i t charge; μ* is a p e r m a n e n t d i p o l e m o m e n t of solvent m o l e c u l e s ; r

k

is t h e distance b e t w e e n t h e d i p o l e a n d t h e acceptor;

is t h e distance b e t w e e n the d i p o l e a n d the donor; r

{j

is t h e distance

b e t w e e n t h e t w o d i p o l e s ; t h e v a r i a b l e r is t h e distance b e t w e e n t h e d o n o r a n d acceptor; e a n d e are e l e c t r o n i c energies o f reactants a n d p r o d u c t s , r

r e s p e c t i v e l y ; E (r) T

eX

p

a n d £ i ( r ) are e l e c t r o n i c energies o f reactants a n d p r o d e

p

ucts w i t h C o u l o m b i n t e r a c t i o n i n v a c u u m , r e s p e c t i v e l y ; V ^ / M

m

^ ^md ( )


P

r

4.

KAKITANI ET AL.

Energy-Gap

Laws of Electron-Transfer

49

Reactions

are the r e s i d u a l c h a r g e - d i p o l e a n d d i p o l e - d i p o l e interactions arising from t h e e l e c t r o n i c p o l a r i z a t i o n o f solute a n d solvent m o l e c u l e s ; a n d V ~ is t h e c o r e - c o r e i n t e r a c t i o n a m o n g solute a n d solvent molecules. T h e explicit forms °f V (r), V ( r ) , a n d V°~° are not necessary i n the p r e s e n t calculations. T h e reaction coordinate o p e r a t o r / i s d e f i n e d (II) as c

i n d

r

i n d

c

p

/ = H'(r) G i b b s e n e r g y curves G (x;r) T

HP(r)

a n d G (x;r)

(8)

as a f u n c t i o n of the r e a c t i o n

p


coordinate χ i n the i n i t i a l a n d final states are w r i t t e n as

e

- G) a

+

(26)

-^-

Z

Similarly,

C/P(r) =

G?(r) -

T h e G i b b s e n e r g y gap -AG(r) -AG(r) =

GP(OO) +

(27)

is g i v e n b y

- A G + l/ (r) 0

r

u»(r)

(28)

w h e r e - A G is t h e standard G i b b s e n e r g y gap, e q u i v a l e n t to -AG(

Electron Transfer in Inorganic, Organic, and Biological Systems

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