2 Basic Electron-Transfer Theory James R. Bolton and Mary D. Archer Downloaded by PENNSYLVANIA STATE UNIV on July 31, 2012 | http://pubs.acs.org Publication Date: May 5, 1991 | doi: 10.1021/ba-1991-0228.ch002
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Photochemistry Unit, Department of Chemistry, University of Western Ontario, London, Ontario Ν6Α 5B7, Canada Newnham College, Cambridge, England CB3 9DF
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This chapter provides an introduction to basic electron-transfer the ory. The classical Marcus theory is developed, and the reorganization energy is defined. The difference between adiabatic and nonadiabatic electron-transfer reactions is explained. Quantum mechanical theo ries of electron transfer are outlined for nonadiabatic reactions with particular application to the Marcus inverted region. Finally, the effect of solvent dynamics is examined.
THE
B A S I C S O F E L E C T R O N - T R A N S F E R T H E O R Y are p r e s e n t e d i n this c h a p t e r
so that the authors of subsequent chapters can refer to it for the f u n d a m e n t a l equations a n d n o m e n c l a t u r e . It s h o u l d also serve as a t u t o r i a l for those w h o are not familiar w i t h the basic theory, a l t h o u g h this is a o n l y a b r i e f o u t l i n e . B y far the most successful theory of e l e c t r o n transfer ( E T ) is that i n t r o d u c e d a n d d e v e l o p e d b y M a r c u s (1-5); thus, this o u t l i n e w i l l d e a l almost exclusively w i t h a s u m m a r y of that t h e o r y a n d the i m p o r t a n t equations d e r i v e d t h e r e f r o m . H u s h (6) d e v e l o p e d a theory s i m i l a r to that of M a r c u s , based o n concepts i n v o l v e d i n E T at electrode surfaces; h o w e v e r , H u s h ' s t h e o r y does not p r e d i c t the i n v e r t e d r e g i o n (vide infra). C o m p r e h e n s i v e r e v i e w s b y N e w t o n a n d S u t i n (7) a n d M a r c u s a n d S u t i n (8, 9) offer a t h o r o u g h d e v e l o p m e n t o f the M a r c u s t h e o r y of e l e c t r o n transfer. U s u a l l y M a r c u s t h e o r y is u s e d for outer-sphere E T reactions b e t w e e n a d o n o r D a n d a n acceptor A . ( F o r c o n v e n i e n c e , w e assume that D a n d A are n e u t r a l m o l e c u l e s . T h e case of c h a r g e d reactants introduces o n l y the possibility of electrostatic effects that can be i n c o r p o r a t e d w i t h little difficulty into the theory.) E i t h e r D o r A m a y b e i n an e x c i t e d state ( D * or A * ) , i n w h i c h case the process is c a l l e d p h o t o i n d u c e d e l e c t r o n transfer ( P E T ) . H o w 0065-2393/91 /0228-0007$06.00/0 © 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.
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E T IN INORGANIC, ORGANIC, A N D BIOLOGICAL SYSTEMS
ever, o t h e r t h a n a change i n the starting-state energies, the p r i n c i p l e s o f electron-transfer t h e o r y a p p l y e q u a l l y w e l l to p h o t o i n d u c e d a n d to g r o u n d state electron-transfer reactions. F o r second-order reactions the E T reaction can b e d i v i d e d into t h r e e steps. I n the first step D a n d A diffuse together to f o r m an o u t e r - s p h e r e precursor complex D|A (rate constant k i n e q l a u s u a l l y approaches the diffusion-controlled limit). a
Downloaded by PENNSYLVANIA STATE UNIV on July 31, 2012 | http://pubs.acs.org Publication Date: May 5, 1991 | doi: 10.1021/ba-1991-0228.ch002
D
+ A f
-
(8)
e
where r and r are the e q u i l i b r i u m b o n d lengths i n the reactant a n d p r o d u c t states, respectively; f is a r e d u c e d force constant for the i t h v i b r a t i o n , a n d the s u m is taken o v e r a l l significant i n t r a m o l e c u l a r v i b r a t i o n s . I n the few cases w h e r e \ values have b e e n calculated, t h e y have b e e n f o u n d to b e fairly s m a l l [ 0 . 1 - 0 . 3 e V ; see B r u n s c h w i g et a l . (12)]; h o w e v e r , i n some inorganic complexes [e.g., C o ( N H ) ] \ can be q u i t e large. R
e q
P
e q
{
i n
3
6
2 + / 3 +
in
T h e outer t e r m X is c a l l e d the solvent reorganization energy because it arises f r o m differences b e t w e e n the orientation a n d p o l a r i z a t i o n of solvent m o l e c u l e s a r o u n d D|A a n d D |A~. I f the s u r r o u n d i n g solvent is t r e a t e d as a d i e l e c t r i c c o n t i n u u m , t h e n it can be s h o w n (1, 13) that o u t
+
j
(E
R
-
Ef p
(9)
dV
w h e r e E a n d E are the electric fields exerted i n vacuo at a distance r f r o m the centers of the reactant a n d p r o d u c t states, r e s p e c t i v e l y ; e a n d e are the o p t i c a l a n d static d i e l e c t r i c constants, r e s p e c t i v e l y , of the s u r r o u n d i n g solvent m e d i u m ( e = n , w h e r e η is the refractive i n d e x of the m e d i u m ) ; R
p
op
op
s
2
e is the p e r m i t t i v i t y of v a c u u m ; a n d the c y c l i c i n t e g r a t i o n is c a r r i e d out over the v o l u m e V. T h e t e r m ( l / e - l/e ) arises because X is the e n e r g y of r e o r g a n i z i n g the solvent molecules a r o u n d the e q u i l i b r i u m D|A c o m p l e x u n t i l t h e y are i n the orientation of the solvent molecules a r o u n d the e q u i l i b r i u m D |A~ c o m p l e x b u t w i t h o u t transfer of the e l e c t r o n . T h i s c o r r e sponds to c h a n g i n g the o r i e n t a t i o n polarization b u t not the n u c l e a r a n d electronic polarization. 0
o p
s
o u t
+
In Electron Transfer in Inorganic, Organic, and Biological Systems; Bolton, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1991.
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E T IN INORGANIC, ORGANIC, A N D BIOLOGICAL SYSTEMS
T h e c o m p u t a t i o n o f the i n t e g r a l of e q 9 r e q u i r e s a specific m o d e l so that a p p r o p r i a t e b o u n d a r y conditions can b e set. M o s t authors have c h o s e n a s p h e r i c a l reagent m o d e l , w h i c h gives o n integration (14)
α
^ _Μ
_!_ _ ±1
+
4ττ€ [_2α
2α
Ό
0
r
Α
D A
μ
.
J |^
ορ
il «U
w h e r e ae is the charge transferred i n the reaction (almost always one e l e c
Downloaded by PENNSYLVANIA STATE UNIV on July 31, 2012 | http://pubs.acs.org Publication Date: May 5, 1991 | doi: 10.1021/ba-1991-0228.ch002
t r o n i c charge); a and r
D A
D
a n d a are the r a d i i of the d o n o r a n d acceptor, r e s p e c t i v e l y ; A
is the center-to-center distance b e t w e e n the d o n o r a n d acceptor.
C a n n o n (13) a n d M a r c u s (15) have also c o n s i d e r e d a m o r e realistic e l l i p s o i d a l m o d e l , b u t it generates rather c o m p l e x equations for X
o u t
. Never
theless, i r r e s p e c t i v e o f the m o d e l chosen, i t is usually possible to approximate ^out
by
Xout »
Β
Γ—
-
(11)
-1
L op
ej
€
w h e r e Β is a s o l v e n t - i n d e p e n d e n t p a r a m e t e r whose value d e p e n d s o n the m o d e l a n d the m o l e c u l a r d i m e n s i o n s . T h e value of X varies f r o m n e a r zero for v e r y n o n p o l a r solvents (for w h i c h € — e ) to 1 . 0 - 1 . 5 e V for polar solvents; thus X is u s u a l l y the d o m i n a n t t e r m i n e q 7. o u t
s
o p
o u t
X is a f u n c t i o n of distance because Β i n e q 11 is a f u n c t i o n o f r [see e q 10]. A l s o , X is slightly t e m p e r a t u r e - d e p e n d e n t , as b o t h e a n d e v a r y w i t h t e m p e r a t u r e . F o r most solvents it is possible to express X = λ TK , w h e r e λ a n d X are e n t h a l p i c a n d entropie c o m p o n e n t s of X , r e s p e c t i v e l y ; this emphasizes that X is a G i b b s energy t e r m . F o r most l i q u i d solvents X does not v a r y b y m o r e t h a n 5 % o v e r a 100 Κ t e m p e r a t u r e range. o u t
D
o u t
op
A
s
o u t
S
Η
s
Η
o u t
o u t
Adiabatic vs. Nonadiabatic Electron-Transfer Reactions.
Two
types o f E T reactions can b e d i s t i n g u i s h e d a c c o r d i n g to the m a g n i t u d e o f the electronic coupling energy Η b e t w e e n the reactant a n d p r o d u c t states (some authors use the s y m b o l V for this term), d e f i n e d b y φ
Η
φ
=
