Parameters of Electron-Transfer Kinetics - ACS Symposium Series

13

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on August 26, 2015 | http://pubs.acs.org Publication Date: September 27, 1982 | doi: 10.1021/bk-1982-0198.ch013

Parameters of Electron-Transfer Kinetics NOEL S. HUSH University of Sydney, Department of Theoretical Chemistry, Sydney, N.S.W. 2006, Australia Over the last two decades, there has been a considerable increase in experimental information about the rates of electron transfer processes, particularly in solution in ionizing solvents. The main features (enthalpies, free energies and entropies of activation) of a large class of these reactions have been broadly understood for about the same length of time. It is interesting to note that the results of very early calculations of outer-sphere reaction rates and their tempera ture dependence frequently agree with recent more detailed estimates. In order to assess the appli cability of different approaches, and thus to discriminate between them, it is desirable to have as much detailed information as possible about the relevant basic parameters. These include, for example, electron coupling or overlap energies, electron-phonon coupling strengths, vibrational frequencies, and spin coupling data. A brief survey is given of the types of experiment which may provide this information. With such data, discussions on matters such as the applicability of Golden Rule perturbation approaches, electronic adiabaticity, the role of nuclear tunnelling, etc., can be put on firmer ground, and some of the outstanding theoretical issues awaiting reso lution are outlined. The theory o f homogeneous e l e c t r o n t r a n s f e r processes, as w e l l as of the c l o s e l y - r e l a t e d e l e c t r o n exchanges w i t h m e t a l l i c e l e c t r o d e s , has been the subject o f c o n s i d e r a b l e study. The p r o p o s a l by Hush and by Marcus t h a t these processes a r e , f o r simple systems, e i t h e r u s u a l l y e l e c t r o n i c a l l y a d i a b a t i c or

0097-6156/82/0198-0301$09.00/0 © 1982 American Chemical Society In Mechanistic Aspects of Inorganic Reactions; Rorabacher, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

MECHANISTIC ASPECTS OF INORGANIC REACTIONS

302

n e a r l y s o , r a t h e r than being s t r o n g l y dominated by e l e c t r o n t u n n e l l i n g p r o b a b i l i t i e s , i s now g e n e r a l l y accepted. For homo geneous t r a n s f e r s i n i o n i z i n g s o l v e n t s a t o r d i n a r y temperatures, the a p p l i c a t i o n o f absolute r e a c t i o n r a t e theory t o exchanges proceeding v i a an outer-sphere mechanism leads t o f a i r l y simple but general expressions f o r t h e f r e e energy, enthalpy and en t r o p y o f a c t i v a t i o n (1-10).


Overview of Progress i n C a l c u l a t i o n of E l e c t r o n - T r a n s f e r Rates A f t e r more than two decades o f i n v e s t i g a t i o n , i t i s p e r t i nent t o ask t o what extent the p r e d i c t i o n s o f t h i s type o f the o r y , and i t s agreement w i t h experiment, have changed. The sys tem which has r e c e i v e d most study i s the aquo f e r r o u s - f e r r i c outer-sphere symmetrical exchange: Fe2+

+

*Fe3+

}

Fe3+ + *Fe2+

In Table I are l i s t e d the f i r s t c a l c u l a t i o n s (1961:8) of the enthalpy, entropy and f r e e energy of a c t i v a t i o n a t 298K. These a r e t o be compared w i t h the v e r y recent (1980) c a l c u l a t i o n s of Newton (11).

Table I 3+ 2+ K i n e t i c Parameters f o r Fe /Fe Self-Exchange

i n Water a t 298K.

[ U n i t s : k c a l mole * o r c a l deg * mole ( F i r s t v a l u e s l i s t e d from Hush (JD > values i n parentheses from Newton ( 1 1 ) ) .

AG*

AS*

AH*

c o n t r i b u t i o n from i n t r a m o l e c u l a r modes

5.0

(7.7)

5.0

(6.4)

0 (-4.1)

c o n t r i b u t i o n from medium modes

7.3

(5.6)

7.3

(5.8)

0 (0.6)

total

20.3

Experiment (11)

An

interesting

(19.7)

(10.4)

-10

-19

feature

10.8

of this

comparison

-32 (-31.3) >-30

i s that the

In Mechanistic Aspects of Inorganic Reactions; Rorabacher, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

13.

HUSH

Parameters of Electron-Transfer Kinetics

303


1

v a r i a t i o n i n the c o n t r i b u t i o n of ' i n n e r and 'outer' terms t o the t o t a l values i s not l a r g e , and the t o t a l values are very c l o s e indeed and a l s o i n reasonable agreement w i t h experiment. Now there are q u i t e a number of d i f f e r e n c e s i n d e t a i l be tween these c a l c u l a t i o n s . The more recent values from Newton, u n l i k e the f i r s t , i n c l u d e i n t e r a l i a allowance f o r departures from e l e c t r o n i c and n u c l e a r a d i a b a t i c i t y . There are a l s o s i g n i f i c a n t conceptual d i f f e r e n c e s between the assumed models. At f i r s t s i g h t , the o v e r a l l agreement of c a l c u l a t i o n s two decades apart might be taken to i n d i c a t e t h a t the models used i n such c a l c u l a t i o n s are too approximate f o r i t t o be p o s s i b l e t o d i s t i n g u i s h between them. I do not, however, b e l i e v e t h a t t h i s i s so. I t h i n k t h a t we can be reasonably c o n f i d e n t t h a t the gen e r a l approach i s c o r r e c t , but the challenge now i s , indeed, to f i n d r e l i a b l e means of t e s t i n g the d e t a i l e d assumptions. Contrasts i n the P r e d i c t e d Free Energy Dependence of Rates

Reaction

A v e r y s i m i l a r s i t u a t i o n e x i s t s w i t h respect t o c r o s s - r e a c t i o n s . We have known f o r a long time (3, 5, 8) t h a t the abso l u t e r e a c t i o n r a t e approach leads t o the p r e d i c t i o n t h a t the f r e e energy of a c t i v a t i o n of the process 2

a

^+ χΖ-1 b

£ a Z - l ^+ .b Ζ

i s approximately the l i n e a r average of the aa and bb s e l f - e x changes, p l u s h a l f the o v e r a l l f r e e energy, provided t h a t t h i s i s not too l a r g e . In Table I I , the r e s u l t s f o r the f i r s t a c t u a l such c a l c u l a t i o n (5, 8) of the e n e r g e t i c s of a c t i v a t i o n f o r a c r o s s - r e a c t i o n i s shown. This i s the p l u t o n y l / p l u t o n i u m ex change : PuO^

+ Pu

3 +

-> PuO^j + P u

4 +

Table I I 2+ λ+ + 44* K i n e t i c Parameters f o r P u 0 + Pu ·* P u 0 + Pu Cross-exchange 2

2

C a l c u l a t e d from T h e o r e t i c a l Values f o r C o n s t i t u e n t Self-exchange i n Aqueous S o l u t i o n (5, 8)

AH* cal. : exp. :

(for (for

I = 0): I = 1):

4.3 4.8

AS*

-40.4

-45 ±

AG*

0.6

17.7 16.9


(298K)


304


The agreement w i t h experiment i s remarkably good and was g r a t i f y i n g a t the time. However, we would now n a t u r a l l y f e e l impelled t o examine the b a s i c assumptions more c l o s e l y before claiming too much f o r t h i s . I t may be asked why such a r e l a t i v e l y complex system was chosen: the answer i s t h a t , a t that time, i t was the only w e l l - c h a r a c t e r i z e d system f o r which data were a v a i l a b l e . This h i g h l i g h t s the very great advances t h a t have been made i n measurement o f e l e c t r o n t r a n s f e r rates since that time. We now have (due, i n l a r g e p a r t , t o the e x c e l l e n t work of Taube (12), Meyer (13), S u t i n (14, 15) and t h e i r c o l l a b o r a t o r s ) a great deal o f information about both s e l f - and crossexchange reactions on which t o draw f o r comparison w i t h theory. The existence o f t h i s body of data makes the f u r t h e r development of theory imperative and worthwhile. I n what f o l l o w s I s h a l l consider what I t h i n k a r e the major p o i n t s t o be resolved. At the ' c l a s s i c a l l e v e l [that i s , one i n which the f r e quencies o f a c t i v e v i b r a t i o n s can be assumed small r e l a t i v e t o kgT] i t i s u s u a l l y stated [e.g., (16)] t h a t the approaches o f 1

Hush and Marcus y i e l d e s s e n t i a l l y i d e n t i c a l r e s u l t s . This i s true f o r self-exchanges. However, i t i s only true w i t h i n a cer t a i n range of AG^, the o v e r a l l f r e e energy, f o r cross-exchanges. The energy o f a chemical system i s a f u n c t i o n a l o f the f i r s t order e l e c t r o n d e n s i t y [ c f . , (17, 18)]. The o r i g i n a l Hush ap proach employs a simple one-electron d e n s i t y f u n c t i o n a l to c a l c u l a t e the d i f f e r e n c e o f f r e e energy between the a c t i v a t e d com plex and the 'precursor' complex. Assuming Born-Oppenheimer s e p a r a b i l i t y , the e l e c t r o n i c wavefunction ψ transferring electron i s : +

,

Ψ (£ »£ 1

2

) =

V-l'V-^l

where r^ ,Γ^ are e l e c t r o n coordinates

+

(:

(

,

+

0^> _2 ~^

r

,

,

)Φ

o

f

t

n

e

2 -1 -2 - 2

f o r centers

1 and 2 and R

i s the i n t e r n u c l e a r separation. The e l e c t r o n i c functions are centered on centers 1 and 2 and φ^ and φ^, r e s p e c t i v e l y . The amplitudes follows :

and

i n each s t a t e o f the r e a c t i o n path are as


13.

HUSH

Parameters of Electron-Transfer

305

Kinetics

state

C

initial

0


precursor c o l l i s i o n complex

1

IP

C

a c t i v a t e d complex successor c o l l i s i o n complex

C

final

1

2

ls

C

2P

C

2*

C

2s

0

The p r o b a b i l i t y d e n s i t y f o r the t r a n s f e r r i n g e l e c t r o n w i t h the acceptor center 1 i n s t a t e i i s then: i c

u

l

associated

= λ.

2

(1)

I f we w r i t e the change o f environmental energy ( i . e . , i n n e r sphere p l u s medium c o n t r i b u t i o n s ) accompanying a v e r t i c a l t r a n s i t i o n from p r e c u r s o r complex t o the upper p o t e n t i a l surface w i t h e l e c t r o n i c wavefunction Φ" = ε (Γ ,Γ ,Κ)φ 2

1

2

1

- C (r ,r ,R)* 1

1

2

(2)

2

as χ (the " r e o r g a n i z a t i o n energy"), the formal t r a n s i t i o n - s t a t e e l e c t r o n d e n s i t i e s , λ*, and a c t i v a t i o n e n e r g i e s , AG*', a r e as f o l l o w s , where AG^' i s the o v e r a l l f r e e energy i n the range p-»s:

àG /x

λ*

Q

AG*'/Χ

1

>1

Si £-1,

« i

AG^ AG' —-2—) χ

+ ο

(χ + A G ' )

2

4χ ο

T h i s type o f approach does not e x p l i c i t l y i n v o l v e s p e c i f i c energy-nuclear c o n f i g u r a t i o n r e l a t i o n s h i p s [e.g., i n t e r s e c t i n g


306



harmonic h y p e r s u r f a c e s ] . However, i n t h e range i n which t h e r a t i o o f the o v e r a l l f r e e energy, A G ^ ( c o r r e c t e d f o r energies of precursor and successor s t a t e f o r m a t i o n ) , t o the r e o r g a n i z a t i o n energy, χ, i s w i t h i n the l i m i t s ±1, the above p r e d i c t i o n s are e s s e n t i a l l y the same f o r w e a k - i n t e r a c t i o n systems as those reached by the f o r m a l l y non-adiabatic approach o f f i n d i n g t h e i n t e r s e c t i o n p o i n t o f two harmonic energy hypersurfaces (the c l a s s i c a l l i m i t o f t h e saddle p o i n t method). This i s i l l u s t r a t e d i n F i g u r e 1, where a q u a d r a t i c dependence on o v e r a l l f r e e energy, w i t h i n these l i m i t s , i s seen t o be p r e d i c t e d by e i t h e r approach. Thus, i n the range 1 ^ A G ^ / x è -1, i t i s c o r r e c t t o say t h a t the Hush and Marcus approaches y i e l d e s s e n t i a l l y the same p r e d i c t i o n s f o r e n e r g e t i c s o f a c t i v a t i o n o f exchange. This r e s t s u l t i m a t e l y , as shown i n (10), on the f o r m a l l y i d e n t i c a l r o l e s o f the e l e c t r o n d e n s i t y parameter λ* and the Lagrange f a c t o r Im I i n the r e s p e c t i v e approaches. Outside these l i m i t s , however, the two approaches r e s u l t i n d i f f e r e n t p r e d i c t e d k i n e t i c behavior o f both exoenergetic and endoenergetic exchanges. I f the r e a c t i o n i s assumed to proceed c l a s s i c a l l y w i t h an energy b a r r i e r corresponding t o the height of t h e reactant/product hypersurface i n t e r s e c t i o n p o i n t , t h e q u a d r a t i c dependence o f A G * on A G ^ i s p r e d i c t e d t o p e r s i s t both f o r h i g h l y endoenergetic and exoenergetic r e a c t i o n s . The den s i t y f u n c t i o n a l approach, however, p r e d i c t s a c t i v a t i o n f r e e energies o f zero and A G ' f o r A G / X

Parameters of Electron-Transfer Kinetics - ACS Symposium Series

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