Further Developments in Electron Transfer - American Chemical Society

10 Further Developments in Electron Transfer

Downloaded by AUBURN UNIV on April 19, 2017 | http://pubs.acs.org Publication Date: September 27, 1982 | doi: 10.1021/bk-1982-0198.ch010

R. A. MARCUS and PAUL SIDERS California Institute of Technology, Arthur Amos Noyes Laboratory of Chemical Physics, Pasadena, CA 91125

The inverted region in electron transfer re actions i s studied for the reaction of e l e c t r o n i cally-excited ruthenium(II) t r i s - b i p y r i d y l ions with various metal(III) t r i s - b i p y r i d y l complexes. Numerical calculations for the diffusion-reaction equation are summarized for the case where elec tron transfer occurs over a range of distances. Comparison i s made with the experimental data and with a simple approximation. The analysis reveals some of the factors which can cause a flattening of the ln k versus ∆Go curve in the inverted obs

region. Ways of improving the chance of observing the effect are discussed. Some time ago i t was p r e d i c t e d ( 1 , 2) t h a t , i n a s e r i e s o f weak-overlap e l e c t r o n t r a n s f e r r e a c t i o n s , the r a t e would f i r s t increase when AG° was made more negative, and then, when AG° be came v e r y negative, e v e n t u a l l y decrease. Evidence f o r such an i n v e r t e d e f f e c t has been given i n a number o f papers (3-11), but i n many other s t u d i e s the r e a c t i o n r a t e reaches a l i m i t i n g 1

1

v a l u e , r a t h e r than a decreasing v a l u e , when -AG° becomes l a r g e (e.g., (12-18)). P o s s i b l e explanations f o r t h e l a t t e r r e s u l t have been suggested: (a) a l t e r n a t e pathways f o r t h e r e a c t i o n when AG° i s v e r y negative [such as Η-atom t r a n s f e r (19, 2 0 ) , e l e c t r o n i c a l l y - e x c i t e d product s t a t e s ( 1 1 , 2 0 ) , o r , when t h e r e a c t i o n was observed v i a quenching o f f l u o r e s c e n s e , e x c i p l e x formation (21, 2 2 ) ] , (b) quantum mechanical n u c l e a r t u n n e l i n g (20, 23-27), (c) masking by d i f f u s i o n , and (d) r e d u c t i o n o f t h e i n v e r t e d e f f e c t [by e l e c t r o n t r a n s f e r over a d i s t a n c e ( 1 9 ) ] . Quantum mechanical t u n n e l i n g reduces the magnitude o f t h e p r e d i c t e d e f f e c t b u t does n o t e l i m i n a t e i t i n weak-overlap sys-

0097-6156/82/0198-0235$06.00/0 © 1982 American Chemical Society Rorabacher and Endicott; Mechanistic Aspects of Inorganic Reactions ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

MECHANISTIC ASPECTS OF INORGANIC REACTIONS

236

terns, as one sees, f o r example, i n some r e c e n t c a l c u l a t i o n s f o r an a c t u a l experimental system (20). Moreover, t h e r e i s a 1:1 correspondence between t h e quantum m e c h a n i c a l l y c a l c u l a t e d charge t r a n s f e r spectrum (emission o r a b s o r p t i o n vs hv) f o r a weak overlap redox system and t h e p l o t (eq 8 and 9 g i v e n l a t e r ) of k versus t h e energy o f r e a c t i o n , ΔΕ ( 2 5 ) , and hence i n a a c t

0

0

series of reactions o f g i v e n A S , versus -AG . Here, k i s t h e a c t i v a t i o n - c o n t r o l l e d quantum m e c h a n i c a l l y c a l c u l a t e d r a t e constant. Thus, the well-known e x i s t e n c e o f a maxi mum i n the charge t r a n s f e r v s wavelength spectrum i m p l i e s t h a t there w i l l be a maximum i n the I n k vs -AG p l o t when the a c t


f i c t

e l e c t r o n t r a n s f e r i s a weak-overlap r e a c t i o n . T h i s correspond ence removes any q u e s t i o n t h a t n u c l e a r t u n n e l i n g would e l i m i n a t e the i n v e r s i o n , s i n c e t h a t t u n n e l i n g occurs t o t h e same e x t e n t i n both the charge t r a n s f e r spectrum and the k vs -AG a c t

p l o t s , and the former has a well-known maximum. I t a l s o r e moves any argument t h a t l a r g e a n h a r m o n i c i t i e s i n p r a c t i c e eliminate the e f f e c t : t h e correspondence a p p l i e s regard l e s s o f whether the v i b r a t i o n s a r e harmonic o r anharmonic, as long as the e l e c t r o n t r a n s f e r i s a weak-overlap one. (The e f f e c t s o f having a v e r y s t r o n g - o v e r l a p e l e c t r o n t r a n s f e r r e main t o be i n v e s t i g a t e d . ) I n a recent paper, an approximate c a l c u l a t i o n was made o f e f f e c t s (b) t o (d) above ( 1 9 ) , u s i n g an approximate a n a l y t i c a l s o l u t i o n f o r the d i f f u s i o n problem, f o r the case where the r e a c t i o n occurs r e a d i l y over a s h o r t range o f s e p a r a t i o n d i s t a n c e s of the r e a c t a n t s . I n the p r e s e n t r e p o r t , we summarize the r e s u l t s o f our recent c a l c u l a t i o n s on a numerical s o l u t i o n o f the same problem. A more complete d e s c r i p t i o n i s g i v e n elsewhere (28). One a d d i t i o n a l m o d i f i c a t i o n made here t o (19) i s t o en sure t h a t the c u r r e n t a v a i l a b l e r a t e constant data a t AG = 0 (Appendix) a r e s a t i s f i e d . 0

Theory The d i f f u s i o n - r e a c t i o n e q u a t i o n f o r the p a i r d i s t r i b u t i o n f u n c t i o n g ( r , t ) o f the r e a c t a n t s , which r e a c t w i t h a r a t e con s t a n t which a t any r i s k ( r ) , i s g i v e n by (29-32)

ΜΕΛΙ 3t

=

l r

îifîV .

k(r)e(t>t)

(1)

Br

where i s the inward r a d i a l f l u x d e n s i t y (per u n i t concentra t i o n ) due t o d i f f u s i o n and t o any f o r c e d motion a r i s i n g from an i n t e r a c t i o n p o t e n t i a l energy, U ( r ) , assumed t o depend o n l y on

Rorabacher and Endicott; Mechanistic Aspects of Inorganic Reactions ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

10. the

MARCUS AND

separation

J

Developments

siDERS

distance

r

+

^

in Electron

Transfer

r . The magnitude o f J

237

f

i s g i v e n by

kgT d r

-U/k^T _ U/k^T De g | (ge ) B

Ξ

Έ

(2)

where D i s t h e sum o f t h e d i f f u s i o n constants o f t h e two r e actants . The observed r a t e constant, k , , a t time t i s then g i v e n by (31, 33) Downloaded by AUBURN UNIV on April 19, 2017 | http://pubs.acs.org Publication Date: September 27, 1982 | doi: 10.1021/bk-1982-0198.ch010

0

k

obs

=

C k(r)g(r,t)47ir

2

D

S

dr

(3)

The s t e a d y - s t a t e s o l u t i o n t o eq 1 s a t i s f i e s 8g/3t = 0, i . e . , i t satisfies 2

2

( l / r ) d ( r J ) / d r = k(r)g(r)

(4)

r

For t h e experimental c o n d i t i o n s i n v e s t i g a t e d thus f a r , t h e s t e a d y - s t a t e s o l u t i o n i s an e x c e l l e n t approximation t o t h e s o l u t i o n o f eq 1 and we consider t h i s case. However, i n proposing some experiments i n the picosecond regime t o enhance t h e chance of observing the i n v e r t e d e f f e c t , we consider t h e time-dependent equation 1. The r a t e constant k ( r ) i s t y p i c a l l y assumed t o depend expo n e n t i a l l y on r , v a r y i n g as exp(-orr). T h e o r e t i c a l estimates have been made f o r Of o f 1.44 8 * when there i s i n t e r v e n i n g m a t e r i a l between t h e r e a c t a n t s (34), and 2.6 8 when there i s n o t (35). A recent c a l c u l a t i o n f o r the hexaaquoiron self-exchange r e a c t i o n 1

y i e l d e d a = 1.8 X " (36). E x p e r i m e n t a l l y , t h e value i n f e r r e d i n d i r e c t l y f o r an e l e c t r o n t r a n s f e r between aromatic systems i n 1

r i g i d media i s about 1.1 8 (37). These values o f α a r e s u f f i c i e n t l y l a r g e t h a t k ( r ) f a l l s o f f r a p i d l y w i t h r . When t h i s " r e a c t i o n d i s t a n c e " i s s m a l l r e l a t i v e t o t h e d i s t a n c e over which t h e f u n c t i o n h ( r ) = g exp (U/kgT) changes s i g n i f i c a n t l y , i . e . , over which ( h ( r ) - η(σ))/ 1

(h(») - η(σ)) becomes a p p r e c i a b l e , one can i n t r o d u c e an a p p r o x i mate a n a l y t i c s o l u t i o n t o eq 4 (28, 38, 3 9 ) : 1

-t

1

(5) k

obs

k

act

k

diff

where, i n t h e present case, we have (from eq 3 w i t h g ( r ) = 0 f o r r < σ)


MECHANISTIC ASPECTS O F INORGANIC REACTIONS

238

-U/k Τ k

act

=

k ( r ) e

B

-C

4 7 l r 2 d r

( 6 )

and where (40)


k

( 7 )

diff =

r

Equation 5 was a c t u a l l y derived f o r the case where r e a c t i o n occurs a t some contact d i s t a n c e r = σ. A d e r i v a t i o n o f eq 5 f o r the present case o f a volume d i s t r i b u t e d r a t e constant k ( r ) i s approximate and i s given elsewhere (28). For k ( r ) we s h a l l assume a t f i r s t , as i n (19), t h a t the r e a c t i o n i s a d i a b a t i c a t the d i s t a n c e o f c l o s e s t approach, r = σ, and t h a t i t i s j o i n e d there t o the nonadiabatic s o l u t i o n which v a r i e s as e x p ( - a r ) . The a d i a b a t i c and nonadiabatic solutions can be j o i n e d smoothly. F o r example, one could t r y t o gener a l i z e t o the present multi-dimensional p o t e n t i a l energy sur f a c e s , a Landau-Zener type treatment ( 4 1 ) . F o r s i m p l i c i t y , however, we w i l l j o i n the a d i a b a t i c and nonadiabatic expressions at r = σ. We subsequently consider another approximation i n which the r e a c t i o n i s t r e a t e d as being nonadiabatic even a t r = σ. The well-known p e r t u r b a t i o n theory expression f o r the non-adiabatic r a t e constant i s given by (25, 42-45) k(r) * ψ

|V(r)|

2

(F.C.)

(8)

where (F.C.) i s the Franck-Condon f a c t o r and V ( r ) i s the e l e c t r o n i c matrix element f o r the e l e c t r o n t r a n s f e r . (F.C.) i s given by -E /k Τ ( F . C . ) = i Σ. e || Ô(E - Ε. + ΔΕ) (9) 1

B

2

f

f

where i and f denote i n i t i a l and f i n a l (reactants* and pro d u c t s ) nuclear c o n f i g u r a t i o n s t a t e s , i n c l u d i n g those o f the s o l v e n t ; AE i s the energy o f r e a c t i o n ; and Q i s I^exp(-E^/kgT). 1

The solvent w i l l be t r e a t e d c l a s s i c a l l y (1) t o avoid the quantum harmonic o s c i l l a t o r treatment o f the p o l a r s o l v e n t which i s sometimes used.

(The l a t t e r y i e l d s a l a r g e e r r o r f o r AS° when

0

AS i s large (46)). The c o n t r i b u t i o n o f the p o l a r s o l v e n t t o the Franck-Condon f a c t o r i s (42, c f . 1) ( F

C

- - S o l v e n t = (^

out

k T)^expl-(AG°" B

+

0

(10)

where AG = AG° + E^ - EY and the s u p e r s c r i p t ν denotes ( i n n e r s h e l l ) v i b r a t i o n a l energy.


10.

for

Developments in Electron

M A R C U S A N D siDERS

Transfer

239

The matching o f the a d i a b a t i c and n o n a d i a b a t i c expressions k ( r ) a t r = σ y i e l d s a value f o r V(a) given by (28) ψ

2

|V(a)| (4n\k T)^ ~ 10 B

1 3

s"

1

(11)

and, f o r a r e o r g a n i z a t i o n parameter λ o f about 70 kJ/mol, y i e l d s |V(a)| - 0.023 eV. This value and


|V(r)|

2

2

= | V ( a ) | e x p [ - o ( r - σ)]

(12)

were introduced i n t o eq 8 as our f i r s t approximation t o V ( r ) . The s e r i e s o f e l e c t r o n t r a n s f e r r e a c t i o n s (14) f o r which we c a l c u l a t e d r a t e constants i n v o l v e quenching o f the lowest ex2+ c i t e d e l e c t r o n i c s t a t e o f Ru(bpy)^ . This * R u ( I I ) s t a t e i s a m e t a l - t o - l i g a n d c h a r g e - t r a n s f e r s t a t e (47, 48) i n which an ex cess e l e c t r o n appears t o be l o c a l i z e d on one o f the b i p y r i d y l l i g a n d s ( 4 9 ) , and t h i s e l e c t r o n may be t r a n s f e r r e d t o a metalcentered o r b i t a l on the o x i d a n t , a t l e a s t when an u n e x c i t e d o x i dant i s formed. A c a l c u l a t i o n o f the d i s t a n c e dependence o f V ( r ) f o r t h i s p a r t i c u l a r t r a n s f e r would be d e s i r a b l e , but l a c k ing t h a t the simple e x p o n e n t i a l form i n d i c a t e d i n eq 12 has been used i n s t e a d . The a c t u a l numerical i n t e g r a t i o n o f eqs 2 and 4 was per formed by c o n v e r t i n g eq 4 t o a p a i r o f o r d i n a r y d i f f e r e n t i a l equations, then u s i n g a standard i n t e g r a t i o n r o u t i n e (50) f o r i n t e g r a t i n g the l a t t e r , i n t e g r a t i n g outward from r = σ t o l a r g e r u n t i l g ( r ) had i t s c o r r e c t f u n c t i o n a l value a t l a r g e r , g ( r ) ~ 1 - c / r where c i s a constant. (This f u n c t i o n a l form i s the s o l u t i o n of eqs 2 and 4 a t r l a r g e enough t h a t k ( r ) = U ( r ) = 0 and f o r U v a n i s h i n g more r a p i d l y than 1/r.) Because g(a) was unknown t o a m u l t i p l i c a t i v e constant i n i t i a l l y , we a c t u a l l y performed the i n t e g r a t i o n f o r a f u n c t i o n G(r) = g ( r ) c ^ , w i t h c^ unknown and w i t h a preassigned v a l u e f o r G ( r ) a t r = σ. The terms c^ and c could be determined from the numerical values o f G a t l a r g e r , and then g ( r ) = G{n)/Cy

The value o f k

Q b g

was

c a l c u l a t e d from the t o t a l f l u x a t r = »: 2

k , = 4ztD l i m ( r ^ ) obs ^ dr' v

= 4/tDc

(13)

Results C a l c u l a t i o n s were Creutz and S u t i n (9)

performed

f o r the system

s t u d i e d by


240

M E C H A N I S T I C ASPECTS O F INORGANIC REACTIONS

*Ru(II)bpy

+ M(III)bpy

3

-> R u ( I I I ) b p y

3

3

+ M(II)bpy

(14)

3

where the bpy* s are v a r i o u s b i p y r i d y l s , M i s one o f s e v e r a l metals, and t h e a s t e r i s k denotes an e l e c t r o n i c a l l y - e x c i t e d mole c u l e . The q u e s t i o n we address i s how, f o r a model which has the "experimental" (Appendix) (k

d i f f

r a t e constant a t AG° = 0 ( k

and the observed

~ 3.5 χ 10

M

s

o b g

~ 4 χ 10

diffusion-limited

rate

8

1

1

M" s" ) constant

) ( 9 ) , do the values p r e d i c t e d f o r k

o b g


f

a t q u i t e negative AG s compare w i t h those c a l c u l a t e d from eq 5 and w i t h the experimental r e s u l t s ? I s t h e e f f e c t o f e l e c t r o n t r a n s f e r over a range o f d i s t a n c e s s u f f i c i e n t l y l a r g e t o e x p l a i n the observed r e s u l t s ( i . e . , v e r y l i t t l e f a l l - o f f o f r a t e con stant with increasing -AG s)? 0f

We use a λ^ 1300

cm"

1

o f 15.5 kJ/mol a s s o c i a t e d w i t h a frequency o f

η

(20), and \

Q u t

o f 54 kJ/mol a t r = σ (51).

A l l cal

c u l a t i o n s were performed w i t h Τ = 298K. The dependence o f \ on r (2) i s i n c o r p o r a t e d i n the c a l c u l a t i o n . An e q u i l i b r i u m Debye-Huckel e x p r e s s i o n f o r the ion-atmosphere-shielded Coulomb i c r e p u l s i o n o f the r e a c t a n t s i s assumed (52, 5 3 ) , given by Q u t

2 U

(

r

)

" E F ~ (1 + —

Ka)

e

f o r t h e case where t h e two r e a c t a n t s have t h e same r a d i u s . Here, κ i s the r e c i p r o c a l o f the Debye-Huckel screening l e n g t h , ε i s the s t a t i c d i e l e c t r i c constant, the z.e„ values are the i o n i c ' ι ο charges o f the r e a c t a n t s , and a i s the d i s t a n c e o f c l o s e s t approach o f the ions i n the i o n atmosphere t o a r e a c t a n t i o n . The d i s t a n c e a i s r . + r , where r . i s the r a d i u s o f a r e a c t a n t ι a' ι i o n and r i s the r a d i u s o f the p r i n c i p a l i o n o f opposite s i g n &

i n the i o n i c atmosphere.

When r . è r , a l i e s between 2 r . and ι a* ι r . , b e i n g 2 r . when r . = r and being r . when r = 0 . Using t h e ι χ ι a ι a c u r r e n t approximate r a d i i we s h a l l , f o r concreteness, take a = 3σ/4. ( I n eq 15 the r e a c t a n t s are assumed t o have the same radius. A more general e x p r e s s i o n than eq 15 i s c i t e d i n r e f . 2 8 ) . A t the p r e v a i l i n g i o n i c s t r e n g t h o f about 0.52 M, r

κ i s about 4.2. 8. Because o f t h i s l a r g e i o n i c s t r e n g t h , U ( r ) i s q u i t e s m a l l , even a t r = σ, 1

Using α = 1.5 J T

1

and, a t f i r s t , V(o) = 0.023 eV, k

AG° = 0 i s found t o be 1.2 χ 1 0

1 0

M^s"

1

a c t

at

which i s s u b s t a n t i a l l y


10.

MARCUS AND SIDERS


241

Transfer

h i g h e r than the c u r r e n t experimental value (Appendix) o f ca 4 χ 8 -1 -1 10 M s Assuming the v a l i d i t y o f the l a t t e r , e i t h e r V(a) i s l e s s than 0.023 eV, i . e . , the r e a c t i o n i s not a d i a b a t i c a t the contact d i s t a n c e r = σ, o r λ i s h i g h e r than estimated, o r eq 15 underestimates U ( r ) . We c o n s i d e r f i r s t u s i n g a d i f f e r e n t V ( a ) , namely, 0.0045 eV, which y i e l d s the c u r r e n t "experimental" r a t e constant a t AG = 0. (The same f i n a l r e s u l t s f o r the I n k ^ g vs 0

AG

p l o t would be o b t a i n e d , e s s e n t i a l l y , i f one used i n s t e a d a 0

d i f f e r e n t ϋ(σ), as long as there i s agreement o f k a t AG = 0.) The numerical s o l u t i o n o f eq 4 and the r a t e constant data 9 -1 -1 of F i g u r e 1 agree a t the data's maximum (~3.5 χ 10 M s ) when -6 2 ™1 one chooses 3.0 χ 10 cm s f o r t h e sum o f t h e D's o f the two reactants. T h i s D i s somewhat near those estimated r a t h e r i n d i r e c t l y ( e l e c t r o c h e m i c a l l y ) f o r the i n d i v i d u a l D s o f f e r r i c and f e r r o u s phenanthroline complexes (~1.9 χ 10 ^ and 3.7 χ 10 ^ 2 -1 cm s , r e s p e c t i v e l y ) ( 5 4 ) . Since r e a c t i o n may a l s o y i e l d e l e c t r o n i c a l l y - e x c i t e d pro ducts when AG° i s s u f f i c i e n t l y n e g a t i v e , we i n c l u d e t h i s r e a c t i o n , as we d i d i n (20). The mean e x c i t a t i o n energy used f o r the formation o f the e l e c t r o n i c a l l y - e x c i t e d R u ( I I I ) product i s 1.76 eV (20). As has been e x p l a i n e d elsewhere (20, 2 8 ) , the formation o f the other p o s s i b l e e l e c t r o n i c a l l y e x c i t e d products i s , i n most cases a t l e a s t , l e s s probable. The same V ( r ) was 3+ used f o r formation o f e l e c t r o n i c a l l y e x c i t e d Ru(bpy)^ as f o r formation o f other products because the d e t a i l e d i n f o r m a t i o n necessary t o make a d i s t i n c t estimate f o r V ( r ) was l a c k i n g . We f i r s t compare the present numerical r e s u l t s f o r the s o l u t i o n o f the s t e a d y - s t a t e eqs 3 and 4 w i t h the approximate s o l u t i o n g i v e n by eqs 5, 6 and the experimental v a l u e f o r k ^ i f f * The r e s u l t s agreed t o about three percent when AG° was v a r i e d from +0.6 t o -3.0 eV. The experimental value f o r k ^ and eq 7 imply a value o f D = 3.5 χ 10 ^ cm s , compared w i t h t h e 3.0 χ -6 2 -1 10 cm s found when eqs 3 and 4 were s o l v e d . Had the same D been used f o r both the exact (eqs 3, 4) and the approximate (eq 5) s o l u t i o n s , t h e i r agreement f o r the r a t e constants would have been about 10% i n s t e a d o f 3%, which i s s t i l l v e r y c l o s e . The r e s u l t s o f s o l v i n g eqs 3 and 4 a r e next compared w i t h the experimental data i n F i g u r e 1 ( 9 ) , u s i n g V(o) = 0.0045 eV. The s o l i d l i n e r e f e r s t o t h e formation o f ground s t a t e p r o d u c t s , and the dotted l i n e t o the formation o f an e l e c t r o n i c a l l y e x c i t e d R u ( I I I ) product. F o r f u r t h e r comparison w i t h t h e s o l i d


a c t

f

d i f

2

1


MECHANISTIC ASPECTS O F INORGANIC


242

REACTIONS

Figure 1. Calculated and experimental rate constants for Reaction 14 vs. AG°. Key: , t-dependent X t; — * — , fixed \ ; , from Réf. 1 in which reaction occurred only at r = σ; and · · ·, current result (r-dependent X ) for formation of an electronically excited product. ou

out

out


10.



Transfer

243

l i n e , a c a l c u l a t i o n was made w i t h λ ^ h e l d f i x e d (54 k J / mol, ' out ' the v a l u e a t r = σ) and i s g i v e n by the dash-dot l i n e . I n order to o b t a i n agreement w i t h the s o l i d l i n e a t AG = 0 , V(a) was reduced t o 0.0039 eV i n c a l c u l a t i n g the dash-dot l i n e . The dashed l i n e i s the r e s u l t o f a c a l c u l a t i o n (20) i n which r e a c t i o n was t r e a t e d as o c c u r r i n g a d i a b a t i c a l l y , but o n l y a t some contact d i s t a n c e σ, and i n which eq 5 was used, together w i t h the experimental value f o r ^^ff The ^ value used f o r t h i s Q U t

l a s t curve was again 54 kJ/mol, the present ^

Q U t

(a).


In F i g u r e 2 we g i v e a comparison o f the s o l i d l i n e o f F i g u r e 1 w i t h t h a t obtained u s i n g V(a) = 0.023 eV and a l a r g e r λ (λ

t

(a)

= 83 kJ/mol).

A s l i g h t l y s m a l l e r D (2.7 χ 1θ"

6

2

1

cm s" )

was r e q u i r e d t o make the l a t t e r c a l c u l a t i o n y i e l d the e x p e r i 9 mental value o f the maximum observed r a t e constant, 3.5 χ 10 M *s Both curves have the same k , a t AG = 0. obs Discussion 0

The r e s u l t s comparing the exact eqs 3 and 4 w i t h the ap proximate eqs 5 and 6 show t h a t the l a t t e r p r o v i d e a good ap proximation f o r the present c o n d i t i o n s , a t l e a s t . The r e s u l t s i n F i g u r e 1 show t h a t , t o account f o r the experimental r e s u l t s at very negative AG ' s u s i n g the present value o f ^-

out

(54 kJ/mol), i t i s necessary t o p o s t u l a t e the formation o f e l e c t r o n i c a l l y - e x c i t e d products. T h i s was a l s o the case i n an e a r l i e r r e s u l t ( 2 0 ) . The sum o f the two r a t e constants i n F i g u r e 1 y i e l d s agreement w i t h the data i n F i g u r e 1 t o a f a c t o r of about 2. I f , as f o r the dashed l i n e i n F i g u r e 2, the v a l u e of λ were a c t u a l l y a p p r e c i a b l y l a r g e r , the formation o f ground s t a t e products alone would s u f f i c e t o o b t a i n agreement. ( C l a s s i c a l l y , the maximum i n the k versus AG curve occurs a t Φ act AG = -λ and so i s s h i f t e d t o more negative AG s when ^ ^ i s increased.) Returning t o F i g u r e 1, one sees t h a t h o l d i n g λ fixed at 0

f

Q U

t

i t s v a l u e a t r = σ (dash-dot l i n e ) does not cause a l a r g e d e v i a t i o n from the more c o r r e c t r e s u l t (r-dependent λ , s o l i d l i n e ) t

i n the i n v e r t e d r e g i o n . A s i m i l a r approximation was used, o f course, f o r the dashed l i n e , where a k ( a ) was used i n s t e a d o f a k(r). We a l s o have explored the s o l u t i o n o f the time-dependent eq 1 t o study the p l o t corresponding t o F i g u r e 1 when the obser v a t i o n o f f l u o r e s c e n c e quenching i n r e a c t i o n 14 i s made a t s h o r t times. I n these short-time c a l c u l a t i o n s we have assumed, f o r



244

s i m p l i c i t y , t h a t r e a c t i o n occurs o n l y a t r = σ. ( C a l c u l a t i o n s are planned f o r the case i n which e l e c t r o n t r a n s f e r occurs over a range o f d i s t a n c e . ) R e s u l t s f o r k ^ g i ^ ) gi f ° several a

r

e

v

e

n

r

times i n F i g u r e 3, and curves are a l s o given f o r the formation of e l e c t r o n i c a l l y - e x c i t e d products. The value o f ^ ^ ( t ) i s

obtained as t h e slope a t time t o f a p l o t o f [ M ( I I I ) b p y ln[*Ru(II)bpy J


3

+ 2 3

]

1

v s t . The r e s u l t s show the enhancement o f the

p r e d i c t e d i n v e r s i o n e f f e c t a t s m a l l times, and an experimental study o f t h i s o r r e l a t e d systems a t such times would be de s i r a b l e , and may, i n f a c t , d i s t i n g u i s h between the p o s s i b i l i t i e s c i t e d e a r l i e r t h a t V(o) < 0.023 eV o r t h a t λ > (15.5 + 54) kJ/mol; a t s h o r t times there would be a double maximum i n the 0

t o t a l r a t e constant versus A G p l o t i n the f i r s t case and a s i n g l e maximum i n the second. The d e t a i l s o f these short-time c a l c u l a t i o n s , made f o r the case t h a t U ( r ) = 0, are g i v e n elsewhere (28). Searching f o r the i n v e r t e d e f f e c t i n u n i m o l e c u l a r systems ( r e a c t a n t s l i n k e d t o each other) would a l s o be v e r y d e s i r a b l e s i n c e t h e i r r a t e s would not be d i f f u s i o n l i m i t e d . Appendix.

'Experimental

1

0

Rate Constant a t A G = 0

The self-exchange r a t e constant f o r r e a c t i o n 14, when M i s Ru and when an e x c i t e d R u ( I I ) product i s formed, has been e s t i '8 -1 -1 mated (55) t o be about 10 M s The self-exchange r a t e con s t a n t f o r r e a c t i o n 14, when M i s Ru and when the products and r e a c t a n t s are i n t h e i r ground e l e c t r o n i c s t a t e s , has been e s t i 9 -1 -1 mated (56) t o be 1.2 χ 10 M s , which i s the observed r a t e 2+ 3+ constant f o r the o x i d a t i o n o f Ru(bpy)^ by Ru(phen)^, f o r which AG° ~ 0.01 eV. Corrected f o r d i f f u s i o n (eq 5 ) , t h e k f o r the 9 -1 -1 l a t t e r i s 2 χ 10 M s . S u t i n (57) has noted t h a t the c r o s s r e l a t i o n ( 1 , 2) should be a p p l i c a b l e t o a n o n a d i a b a t i c e l e c t r o n t r a n s f e r i f the e l e c t r o n i c m a t r i x element, V ( r ) , f o r the c r o s s r e a c t i o n i s equal t o the geometric mean o f the m a t r i x elements f o r t h e self-exchange r e a c t i o n s . Assuming t h a t t h a t c o n d i t i o n i s approximately s a t i s f i e d , t h e exchange r a t e constant f o r r e a c t i o n 14 when AG° = 0 i s estimated t o be the geometric mean, (20 χ 1 ) ^ χ 1 0 M" s" , i . e . , 4.5 χ 1 0 f f V . Corrected f o r 8 -1-1 d i f f u s i o n u s i n g eq 5, t h i s becomes 4 χ 10 M s , the v a l u e given i n the t e x t . I n t e r e s t i n g l y enough, t h e r a t e constant a t AG = 0 f o r r e a c t i o n 14 when * R u ( I I ) and M ( I I I ) are r e p l a c e d by * C r ( I I I ) and R u ( I I ) , r e s p e c t i v e l y , i s - 2 χ 1 0 M" s" i n 1 M H S 0 ( 1 0 ) . 8

1

1

8

1

o

8

1

1

2


4


10.



Transfer

245

AG°(eV) Figure 2 . Calculated rate constants for Reaction 14 vs. A G ° . Key: from solid line in Figure 1, V(a) = 0.0045 eV, KuM = 54 kJ/mol; V(a) = 0.023 eV and KuM = 83 kJ/mol.

, taken and ,

AG° (eV) Figure 3. Time-dependent calculations of k (t) vs. AG° for various observation times. Key: - · - , 1 ps; - · · - , 5 ps; and · · · , kobefU for formation of an excitedstate Ru(III). ohe


246


Acknowledgment I t i s a pleasure to acknowledge support of t h i s research by a grant from the N a t i o n a l Science Foundation. C o n t r i b u t i o n No. 6494 from the C a l i f o r n i a I n s t i t u t e o f Technology.


Literature Cited 1. Marcus, R. A. Discuss. Faraday Soc. 1960, 29, 21. 2. Marcus, R. A. J. Chem. Phys. 1965, 43, 2654. 3. Allen, A. O.; Gangwer, T. E.; Holroyd, R. A. J. Phys. Chem. 1975, 79, 25. 4. Henglein, A. Can. J. Chem. 1977, 55, 2112. 5. Lipsky, S. J. Chem. Ed. 1981, 58, 93. 6. Frank, A. J.; Gratzel, M.; Henglein, Α.; Janata, E. Ber. Bunsenges. Phys. Chem. 1976, 80, 294. 7. Frank, A. J.; Grätzel, M.; Henglein, Α.; Janata, E. Ber. Bunsenges. Phys. Chem. 1976, 80, 547. 8. Ulstrup, J. "Charge Transfer Processes in Condensed Media, Lecture Notes in Chemistry, No. 10"; Springer-Verlag: New York, 1979; pp. 163-4. 9. Creutz, C.; Sutin, N. J. Am. Chem. Soc. 1977, 99, 241. 10. Brunschwig, B.; Sutin, N. J. Am. Chem. Soc. 1978, 100, 7568. 11. Beitz, J. V.; Miller, J. R. J. Chem. Phys. 1979, 71, 4579. 12. Rehm, D.; Weller, A. Israel J. Chem. 1970, 8, 259. 13. Romashov, L. V.; Kiryukhin, Yu. I.; Bagdasar'yan, Kh. S. Doklady Phys. Chem. (Engl. Transl.) 1976, 230, 961. 14. Scheerer, R.; Grätzel, M. J. Am. Chem. Soc. 1977, 99, 865. 15. Ballardini, R.; Varani, G.; Indelli, M. T.; Scandola, F.; Balzani, V. J. Am. Chem. Soc. 1978, 100, 7219. 16. Balzani, V.; Bolletta, F.; Gandolfi, M. T.; Maestri, M. Top. Curr. Chem. 1978, 75, 1. 17. Eriksen, J.; Foote, C. S. J. Phys Chem. 1978, 82, 2659. 18. Vogelmann, E.; Schreiner, S.; Rauscher, W.; Kramer, Η. Ε. A. Z. Phys. Chem. N. F. 1976, 101, 321. 19. Marcus, R. A. Int. J. Chem. Kin. 1981, 13, 865. 20. Siders, P.; Marcus, R. A. J. Am. Chem. Soc. 1981, 103, 748. 21. Weller, Α.; Zachariasse, K. Chem. Phys. Lett. 1971, 10, 590. 22. Joussot-Dubien, J.; Albrecht, A. C.; Gerischer, H.; Knox, R. S.; Marcus, R. Α.; Schott, M.; Weller, Α.; Willig, F. Life Sci. Res. Rep. 1979, 12, 129. 23. Efrima, S.; Bixon, M. Chem. Phys. Lett. 1974, 25, 34; Chem. Phys. 1976, 13, 447. 24. Van Duyne, R. P.; Fischer, S. F. Chem. Phys. 1974, 5, 183. 25. Ulstrup, J.; Jortner, J. J. J. Chem. Phys. 1975, 63, 4358.



10.

MARCUS AND SIDERS


Transfer

247

26. Marcus, R. Α.; Sutin, N. Inorg. Chem. 1975, 14, 213. 27. Schmickler, W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 307. 28. Marcus, R. Α.; Siders, P. J. Phys. Chem. 1982, 86, 622. 29. Waite, T. R. J. Chem. Phys. 1958, 28, 103. 30. Pilling, M. J . ; Rice, S. A. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1563. 31. Butler, P. R.; Pilling, M. J . ; Rice, S. Α.; Stone, T. J. Can. J. Chem. 1977, 55, 2124. 32. Wilemski, G.; Fixman, M. J. Chem. Phys. 1973, 58, 4009. 33. Keizer, J. J. Phys. Chem. 1981, 85, 940. 34. Hopfield, J. J. Proc. Nat. Acad. Sci., USA 1974, 71, 3640. 35. Jortner, J. J. Chem. Phys. 1976, 64, 4860. 36. Newton, M. D. Int. J. Quant. Chem.: Quant. Chem. Symp. 1980, 14, 363. 37. Alexandrov, I. V.; Khairutdinov, R. F.; Zamaraev, Κ. I. Chem. Phys. 1978, 32, 123. 38. Noyes, R. M. Progr. Reaction Kinetics 1961, 1, 129. 39. Marcus, R. A. Discuss. Faraday Soc. 1960, 29, 129. 40. Debye, P. Trans. Electrochem. Soc. 1942, 82, 265. 41. Brunschwig, B. S.; Logan, J . ; Newton, M. D.; Sutin, N. J. Am. Chem. Soc. 1980, 102, 5798. 42. Kestner, N.; Logan, J . ; Jortner, J. J. Phys. Chem. 1974, 78, 2148. 43. Levich, V. G.; Dogonadze, R. R. Collect. Czech. Chem. Commun. 1961, 26, 193. 44. Levich, V. G. "Physical Chemistry: An Advanced Treatise"; Eyring, H.; Henderson, D.; Jost, W., Editors; Academic Press: New York, 1970; Vol. 9B. 45. Dogonadze, R. R.; Kuznetsov, A. M.; Vorotyntsev, M. A. Phys. Status Solidi Β 1972, 54, 125, 425. 46. Siders, P.; Marcus, R. A. J. Am. Chem. Soc. 1981, 103, 741. 47. Palmer, R. Α.; Piper, T. S. Inorg. Chem. 1966, 5, 864. 48. Lytle, F. E.; Hercules, D. M. J. Am. Chem. Soc. 1969, 91, 253. 49. Sutin, N.; Creutz, C. Adv. Chem. Ser. 1978, No. 168, 1. 50. Gordon, M. K. Sandia Laboratories Report, SAND75-0211. For a discussion of the algorithm, see Shampine, L. F.; Gordon, M. K. "Computer Solution of Ordinary Differential Equations"; Freeman: San Francisco, 1975. 51. Sutin, N. in "Tunneling in Biological Systems"; Chance, B.; DeVault, D. C.; Frauenfelder, H.; Schrieffer, J. R.; Sutin, N.; Eds., Academic Press: New York, 1979; p. 201. 52. Debye, P.; Hückel, E. Physikal Z. 1923, 24, 185. 53. Moore, W. J. "Physical Chemistry"; 4th ed.; Prentice-Hall: Englewood Cliffs, N. J., 1972; eq 10.58. 54. Ruff, I.; Zimonyi, M. Electrochimica Acta 1973, 18, 515.


248


55. Lin, C-T.; Bottcher, W.; Chou, M.; Creutz, C.; Sutin, N. J. Am. Chem. Soc. 1976, 98, 6536. 56. Young, R. C.; Keene, F. R.; Meyer, T. J. J. Am. Chem. Soc. 1977, 99, 2468. 57. Sutin, N. Acc. Chem. Res. 1968, 1, 225. 1982.


RECEIVED April 21,



General Discussion—Further Developments i n Electron Transfer Leader: P. P. Schmidt DR. THOMAS MEYER ( U n i v e r s i t y o f North C a r o l i n a ) : One e x p e r i m e n t a l l y v a l i d approach t o t h i s problem o f the i n v e r t e d r e g i o n begins w i t h a systematic study o f a s e r i e s o f r e l a t e d metal b y p y r i d i n e charge t r a n s f e r e x c i t e d s t a t e s . I n these e x c i t e d s t a t e s there are r u t h e n i u m ( I I I ) or osmium(III) cores bound, i f you w i l l , t o a l i g a n d r a d i c a l anion. By making v a r i a t i o n s i n the o t h e r f o u r l i g a n d s o f these s i x - c o o r d i n a t e com p l e x e s i t i s p o s s i b l e t o v a r y s y s t e m a t i c a l l y the energy gap ( i . e . , the spacing between the upper l e v e l s u r f a c e s ) and t o measure r a d i a t i o n l e s s decay r a t e s from l i f e t i m e and quantum y i e l d measurements. The r e s u l t s show t h a t good l i n e a r c o r r e l a t i o n s e x i s t between Zn and AE i n agreement w i t h the weak v i b r a t i o n a l c o u p l i n g l i m i t e x p r e s s i o n d e r i v e d by Englman and J o r t n e r [Englman, R.; J o r t n e r , J . Molec. Phys. 1970, 18, 145; c f . , C u r t i s , J . C ; B e r n s t e i n , J . S.; Schmehl, R. H. ; Meyer, T. J . Chem. Phys. L e t t . 1981, 81, 4 8 ] . DR. LESLIE DUTTON ( U n i v e r s i t y of P e n n s y l v a n i a ) : I am i n t r i g u e d by the f a c t t h a t people have not taken B e i t z and M i l l e r ' s data more s e r i o u s l y [ B e i t z , J . V.; M i l l e r , J . R. J . Chem. Phys. 1979, 71, 4579]. M i l l e r showed t h a t i t i s pos s i b l e t o f i n d a data s e t f o r s i m i l a r chemicals going two or three orders of magnitude over the hump. Why are those data b e i n g ignored? DR. SIDERS: The data o f B e i t z and M i l l e r are v e r y i n t e r e s t i n g but I f e e l u n c e r t a i n about t h e i r c o r r e c t i n t e r p r e t a t i o n be cause the measurements were f o r t r a n s f e r o f an e l e c t r o n from a s o l v e n t t r a p r a t h e r than from a molecule. A l s o , there's l o t o f s c a t t e r i n the data, although they do seem t o show i n v e r s i o n . F i n a l l y , the data were obtained i n a 2-methyl-tetrahydrofuran g l a s s a t 77°K, which may d i f f e r s i g n i f i c a n t l y from water a t room temperature. DR. MARSHALL NEWTON (Brookhaven N a t i o n a l L a b o r a t o r y ) : I'd l i k e t o ask a q u e s t i o n about H o p f i e l d ' s numbers. The alpha parameter from h i s 1974 paper [ H o p f i e l d , J . J . Proc. N a t l . Acad. S c i . , USA 1974, 71, 3640] was based not on the d i r e c t metal-metal i n t e r a c t i o n but r a t h e r was based on carbon-carbon o v e r l a p because i t was two carbons which were c l o s e s t t o g e t h e r i n h i s e l e c t r o n t r a n s p o r t system. I n c o n t r a s t , Dr. S u t i n gave some d i f f e r e n t numbers based on metal o r b i t a l s . Depending on whether one i s i n t e r e s t e d i n carbon-carbon o v e r l a p between two o r g a n i c r i n g s , or i n d i r e c t metal-metal o v e r l a p , one might or might not opt f o r the H o p f i e l d parameters. However, a t the l e v e l of f u z z i n e s s which we have, i t may not make any d i f f e r ence, I r e a l i z e .



250

DR. SIDERS: The edge atoms on the r i n g s o f the t r i s - b i p y r i d y l complexes t h a t we considered are a long way out from the c e n t r a l metal i o n s . For t h a t reason we thought t h a t an estimate such as H o p f i e l d s , based on a carbon-carbon o v e r l a p , would be more appropriate than one based on a metal-metal overlap. 1

DR. EPHRAIM BUHKS ( U n i v e r s i t y o f Delaware): I would l i k e to mention b r i e f l y some recent work which demonstrates that quantum-mechanical c a l c u l a t i o n s r e a l l y can provide a b a s i s f o r understanding the mechanism o f slow e l e c t r o n exchange i n systems 3+ 2+ such as Co(NH )^ * [Buhks, E.; Bixon, M.; J o r t n e r , J . ; Navon, G. Inorg. Chem. 1979, 18, 2014]. The e l e c t r o n t r a n s f e r r a t e can be represented i n terms o f a product o f the e l e c t r o n exchange matrix element and FranckCondon f a c t o r s ; the l a t t e r takes i n t o account the c o n t r i b u t i o n s of s o l v e n t p o l a r i z a t i o n and i n t r a m o l e c u l a r - v i b r a t i o n a l modes both o f the acceptor and donor i o n s . These f a c t o r s , i n g e n e r a l , i n c o r p o r a t e the c o n t r i b u t i o n o f the frequency change. The de t a i l e d c a l c u l a t i o n , c o n s i d e r i n g 30 v i b r a t i o n a l modes, demon s t r a t e s t h a t the frequency change i s not v e r y important f o r t h i s electron-exchange r e a c t i o n . I t r a t h e r p r o v i d e s some s m a l l f a c t o r o f 10 * o r so. The most important term, which i n c l u d e s the exponential o f the square o f the change i n m e t a l - l i g a n d bond d i s t a n c e s , i s r e s p o n s i b l e f o r t h e e i g h t orders o f magnitude 3+ 2+ r a t i o between Franck-Condon f a c t o r s f o r the Ru(NH^), ' and 3+2+ Co(NH^)^ ' exchange. The other c o n t r i b u t i o n which should be taken i n t o account i s the e l e c t r o n exchange matrix element. 3+ E l e c t r o n exchange between the ground s t a t e s o f CoiNH^), and 2+ Co(NH^)^ i s s p i n forbidden. For t h i s reason, the t r u e e l e c t r o n i c s t a t e s should take i n t o account a combination o f t h e ground s t a t e w i t h e x c i t e d e l e c t r o n i c s t a t e s f o r both C o ( I I ) and Co(III). Thus, i n the c a l c u l a t i o n o f t h e e l e c t r o n exchange matrix -4 element, an a d d i t i o n a l f a c t o r o f 10 appears due to the mixing of the ground s t a t e s w i t h e x c i t e d s t a t e s and t h e i r cross terms. A l t o g e t h e r , t h e o r e t i c a l l y , one can account f o r the 12 orders o f magnitude d i f f e r e n c e i n the r e a c t i v i t y o f the ruthenium and co b a l t couples. An a d d i t i o n a l concern a r i s e s i n regard t o any d i f f e r e n c e s which may e x i s t between the c l a s s i c a l theory and the quantummechanical approach i n t h e c a l c u l a t i o n o f the Franck-Condon f a c t o r s f o r symmetrical exchange r e a c t i o n s . I n f a c t , the d i f ference i s n o t very l a r g e . F o r a frequency o f 400 cm * f o r m e t a l - l i g a n d t o t a l l y symmetric v i b r a t i o n a l modes, one can expect


3


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M A R C U S A N D SIDERS

Developments

in Electron

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only one order o f magnitude d i f f e r e n c e between r a t e constants c a l c u l a t e d u s i n g c l a s s i c a l and quantum-mechanical models f o r systems e x h i b i t i n g such l a r g e changes i n m e t a l - l i g a n d d i s t a n c e 3+ 2+ as e x h i b i t e d by the Co(NH^)^ couple [Buhks, E.; Bixon, M.; J o r t n e r , J . ; Navon, G. J . Phys. Chem. 1981, 85, 3759]. On the other hand, i f one considers the v a r i a t i o n o f a c t i v a t i o n energy w i t h e l e c t r o n i c energy gap, one f i n d s a v e r y l a r g e discrepancy between the quantum-mechanical and c l a s s i c a l approaches f o r v e r y exothermic r e a c t i o n s a t room temperature (see F i g u r e 1 ) . A f u r t h e r e x p l o r a t i o n o f the n u c l e a r t u n n e l i n g phenomena was presented i n the study o f deuterium isotope e f f e c t s on electron-exchange r e a c t i o n s [Buhks, E.; Bixon, M. ; J o r t n e r , J . J . Phys. Chem. 1981, 85, 3763]. The main cause o f the isotope e f f e c t on the r a t e o f e l e c t r o n exchange o r i g i n a t e s from the changes i n the m e t a l - l i g a n d v i b r a t i o n a l frequencies which are induced by a change o f the mass o f the l i g a n d . The e f f e c t a l s o depends on the changes i n the m e t a l - l i g a n d e q u i l i b r i u m c o n f i g u r a t i o n s accompanying e l e c t r o n t r a n s f e r and on temperature. F o r a system c h a r a c t e r i z e d by a s u b s t a n t i a l change i n the metal3+ 2+ l i g a n d d i s t a n c e , such as Co(NH^)^ * , kjj/kp changes from ~1.3 at room temperature t o ~30 a t f a i r l y low temperatures, w h i l e i t i s u n i t y i n the h i g h temperature c l a s s i c a l l i m i t . I n the tem perature range 0°-70°C i t i s expected t h a t Zn(k^/k^) should


9

decrease as Τ I

The deuterium isotope e f f e c t , k^/k^, e x h i b i t s

a maximum f o r the symmetric e l e c t r o n exchange r e a c t i o n s , and decreases f o r a c t i v a t i o n l e s s and b a r r i e r l e s s r e a c t i o n s (see Figure 2 ) . These p r e d i c t i o n s can provide an experimental t e s t o f the mechanism f o r quantum-mechanical t u n n e l i n g e f f e c t s on e l e c t r o n t r a n s f e r processes i n s o l u t i o n and i n glasses over a wide tem perature range.




252

Figure 1. Activation energy of electron-transfer process as a function of electronic energy gap of a reaction. E r = E g + E c is the total reorganization energy where E 8 is the classical solvent reorganization energy and E c is the reorganization energy of an intramolecular mode, = 2k B T, at room temperature. Curve 1 (E = 0) represents a classical case; curve 3 (E = 0) represents quantum effects at room temperature; and curve 2 (E = E c = B /2) represents the interference of the two previous cases. c

B

B

T



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3

-

2

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1

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Transfer

1

2

AE/E

C

Figure 2. Deuterium isotope effect for electron transfer between ammine complexes as a function of the reduced electronic energy gap A E / E where E is the total reorganization energy E,. = E + E . Key for parameters: , ha> /k T C

- 2.0 and E / E = 0; 8

c

8

c

, E / E = 1; and 8

c

r

, E / E = 2. s

c


H

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Further Developments in Electron Transfer - American Chemical Society

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