Formalism for Electron Transfer and Energy Transfer in Bridged Systems

2 Formalism for Electron Transfer and Downloaded by UNIV OF NEW SOUTH WALES on September 3, 2015 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch002

Energy Transfer in Bridged Systems J . R. Reimers and N. S. Hush

1

Department of Theoretical Chemistry, University of Sydney, Sydney, New South Wales 2006, Australia

There is much current interest in the study of electron transfer or energy transfer between centers linked by a bridge, and a number of basic points need clarification in the interpretation of such processes. For example, it is commonly assumed that communication through the bridge can be described in terms of an effective "two— level" model; under what circumstances is this justified? What are the conditions under which a single-parameter rate constant characterizes the process? With a view to providing some information about these and related questions, a formalism applicable to coherent transfer of energy (including electron transfer and hole transfer in cases where the vibrational motions do not need to be explicitly considered) is developed. This formalism yields a completely general algorithm, which, in particular limits, reduces to a generalized form of Fermi's golden rule and of Rabi's rate equation. In so doing, it unifies a number of existing theories.

E

L E C T R O N (OR H O L E ) A N D E N E R G Y T R A N S F E R b e t w e e n a m e t a l d o n o r c e n

ter l i n k e d to a n acceptor b y a b r i d g e is increasingly b e i n g s t u d i e d e x p e r i m e n t a l l y . S u c h systems often i n v o l v e m i x e d - v a l e n t m e t a l states; b i o i n o r g a n i c systems (both naturally o c c u r r i n g ( J , 2) a n d synthetic) are also o f m u c h c u r r e n t interest. D e s i g n o f m o l e c u l a r electronic devices, i n w h i c h t h e r e is also m u c h c u r r e n t interest, frequently involves elements o f this general k i n d . I n m o l e c u l a r electronics, o n e desires to find ways o f transferring electrons u n d e r c o n t r o l l e d conditions b e t w e e n different states o f a d e v i c e . N a t u r e has Address correspondence to this author.

J

0065-2393/90/0226-0027$10.25/0

© 1990 American Chemical Society

In Electron Transfer in Biology and the Solid State; Johnson, M., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

28

E L E C T R O N TRANSFER IN BIOLOGY A N D T H E SOLID STATE

Downloaded by UNIV OF NEW SOUTH WALES on September 3, 2015 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch002

p r o v i d e d models for this process, a n d w e can l e a r n m u c h from s t u d y i n g processes such as photosynthesis ( I , 2), i n w h i c h l i g h t p r o d u c e s an e l e c t r o n i n an excited state (the d o n o r state) that reacts to transfer the e l e c t r o n to a geographically distant state (the acceptor state). I n d e e d , it is possible to manufacture c o m p o u n d s i n w h i c h s u c h transfers o c c u r u n d e r c o n t r o l l e d conditions (3,4), a n d s u c h c o m p o u n d s are candidates for m o l e c u l a r e l e c t r o n i c devices. U s u a l l y , the b r i d g e is c o m p o s e d of organic m a t e r i a l a n d m a y b e e i t h e r a c o n d u c t o r (e.g., a conjugated bridge) o r a n insulator (e.g., a saturated hydrocarbon). S o m e day processes m a y b e d e v e l o p e d b y w h i c h the conductance o f the b r i d g e can b e m o d i f i e d significantly b y an external signal (5,6). I n this chapter w e generate a m o d e l that describes h o w the conductance o f the b r i d g e varies w i t h respect to some f u n d a m e n t a l properties o f the b r i d g e . W e do not c o n s i d e r the q u e s t i o n o f h o w these properties m a y b e m o d i f i e d d y n a m i c a l l y . O u r m o d e l is i n s p i r e d b y the observation that n o r m a l l y i n s u l a t i n g bridges can i n fact b e c o m e excellent conductors (see refs. 3 a n d 4). T h e basic f o r m a l i s m is d e v e l o p e d i n this chapter. P e r t u r b a t i o n e x p a n sions are p e r f o r m e d to describe the kinetics w h e n the b a t h c o u p l i n g is w e a k a n d w h e n it is strong, a n d an accurate i n t e r p o l a t i o n s c h e m e is p r e s e n t e d for the i n t e r m e d i a t e r e g i o n . T h e r e l a t i o n s h i p o f o u r approach to the effective t w o - l e v e l models o f J o a c h i m (39), o f B r o o a n d L a r s s o n (45), a n d o f M c C o n n e l l (24) is s h o w n . A s a n a p p l i c a t i o n , o u r m o d e l is a p p l i e d to t h r e e - l e v e l systems c o n t a i n i n g j u s t one b r i d g e state. T h e e l e c t r o n transfer is s h o w n to b e m u c h faster w h e n the b r i d g e state is resonant w i t h the d o n o r a n d acceptor states than w h e n it is nonresonant. T h i s system is a m o d e l system for e l e c t r o n transfer t h r o u g h bridges. Q u a n t i t a t i v e features of the m o d u l a t i o n of the donor-to-acceptor c u r r e n t are i n d u c e d b y changes made to b r i d g e p r o p e r t i e s . F i n a l l y , a study o f the b r i d g e - l e n g t h d e p e n d e n c e o f the reaction-rate constant is p e r f o r m e d b y u s i n g M c C o n n e l F s m o d e l H a m i l t o n i a n .

Electron-Transfer Theory Inorganic systems have p l a y e d a c r u c i a l role i n the d e v e l o p m e n t o f e l e c t r o n transfer theory. T h i s role is e x e m p l i f i e d i n the w o r k o f T a u b e et a l . (7-14) a n d others (25) o n m i x e d - v a l e n c e b i s r u t h e n i u m complexes. T h i s c h a p t e r presents a f o r m a l i s m for c o n s i d e r i n g electron-transfer p r o b l e m s . T h i s for m a l i s m is a p p l i e d i n later papers (26, 27) to study e l e c t r o n transfer i n these systems. M o r e generally, e l e c t r o n - a n d energy-transfer p h e n o m e n a have b e e n s t u d i e d extensively (28-47) a n d r e v i e w e d (46-52). U s u a l l y , t h r o u g h - b r i d g e studies (45) i m p l i c i t l y i n t r o d u c e the assumption that the k i n e t i c s can b e d e s c r i b e d b y u s i n g an effective t w o - l e v e l m o d e l i n w h i c h the total H a m i l tonian is r e p l a c e d b y an effective d o n o r - a c c e p t o r i n t e r a c t i o n H a m i l t o n i a n . W e generated a m o d e l for the rate constant that is not based o n this as-


2.

REIMERS & H U S H

29

Bridged Systems

s u m p t i o n . O u r m o d e l provides resolutions for a m b i g u i t i e s i n e a r l i e r theories and shows h o w these approaches relate to b o t h each o t h e r a n d f u n d a m e n t a l physics. W e started w i t h the energy-transfer m o d e l d e

T w o - L e v e l Systems.

v e l o p e d b y R o b i n s o n a n d F r o s c h (18-20), w h i c h describes energy decay i n t w o - l e v e l systems a n d i n networks of such systems. E l e c t r o n - t r a n s f e r p r o c esses f o r m a subset of energy-transfer processes a n d feature the transfer o f Downloaded by UNIV OF NEW SOUTH WALES on September 3, 2015 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch002

an e l e c t r o n (or hole) over some distance, usually large c o m p a r e d to the spatial extent o f the d o n o r a n d acceptor states. I n the m o r e g e n e r a l process, the e l e c t r o n c o u l d transfer b e t w e e n d o n o r a n d acceptor states w i t h i n the same r e g i o n o f space a n d thus p r o d u c e n e g l i g i b l e d i p o l e m o m e n t change. T h e language o f this chapter is consistent w i t h the m o r e - g e n e r a l p h e n o m e n o n , a n d so the terms " d o n o r " a n d " a c c e p t o r " always refer to the electronic states i n v o l v e d a n d not to c h e m i c a l species or

fragments.

The

d o n o r a n d acceptor states are not eigenstates o f the c o m p l e t e H a m i l t o n i a n , and thus they are not stationary states. R a t h e r , they evolve i n t i m e , a n d it is this t i m e e v o l u t i o n w i t h w h i c h w e are c o n c e r n e d . D e p e n d e n c e o n the B a t h . R o b i n s o n a n d F r o s c h d e s c r i b e d a large range of p h e n o m e n a w i t h a theory c o n t a i n i n g j u s t two variables: the H a m i l t o n i a n matrix e l e m e n t V c o n n e c t i n g the two l e v e l s , a n d the accep t o r - s o l v e n t c o u p l i n g X of the final state to a b a t h . (These variables w e r e originally (18) n a m e d (3 a n d a / 2 , respectively.) B y some means (e.g., e l e c t r i c a l , t h e r m a l , p h o t o c h e m i c a l , o r mechanical), energy is p l a c e d i n the d o n o r state. A s this d o n o r state is not an eigenstate, it evolves i n t i m e b o t h co h e r e n t l y a n d r e v e r s i b l y , thus transferring e n e r g y to a n d from the acceptor 1 2

state. T h e acceptor state, h o w e v e r , is also c o u p l e d to a v e r y large n u m b e r o f a d d i t i o n a l states, w h i c h are c o l l e c t i v e l y k n o w n as the b a t h . P h y s i c a l l y , the b a t h m a y be a s u r r o u n d i n g fluid, p r o t e i n or crystal, a n d it m a y e v e n i n c l u d e i n t e r n a l m o l e c u l a r degrees o f freedom. Because the b a t h contains a v e r y large n u m b e r o f states, it is h i g h l y i m p r o b a b l e that energy w i l l b e transferred back f r o m the b a t h to the acceptor. Excess e n e r g y is thus lost, p r e v e n t i n g the transfer o f energy back from the acceptor to the donor. T h e final result is that energy is transferred from the d o n o r to the acceptor, a n d a s m a l l a m o u n t of excess energy is also transferred to the b a t h . O u r approach extends the m o d e l of R o b i n s o n a n d F r o s c h to i n c l u d e p r o b l e m s i n w h i c h the d o n o r and acceptor are c o u p l e d t h r o u g h a b r i d g e rather t h a n j u s t c o u p l e d d i r e c t l y (through space). Acceptor—Solvent C o u p l i n g . A n essential feature o f this t h e o r y is that the electron-transfer rate constant is e x p l i c i t l y d e p e n d e n t u p o n the a c c e p t o r - s o l v e n t c o u p l i n g X. T h u s , properties o f the solvent are d i r e c t l y


30



i n c l u d e d . O t h e r solvent p r o p e r t i e s , such as static shifts o f state energies a n d force constants, are i n c l u d e d i n d i r e c t l y i n the use o f s o l v e n t - d e p e n d e n t d o n o r - b r i d g e - a c c e p t o r p o t e n t i a l parameters. I f the solvent is a d i l u t e gas, t h e n t h e c o u p l i n g X is p r o b a b l y related to the m e a n c o l l i s i o n frequency ( " i n t e r n a l " collisions m a y n e e d to be i n c l u d e d here) a n d can thus b e c o m e q u i t e small. T h e donor-to-acceptor rate constant cannot exceed k/fi (fl is the P l a n c k constant). T h u s , i n a d i l u t e gas the electron-transfer rate constant w i l l b e s m a l l , i n d e p e n d e n t o f the nature o f the d o n o r a n d acceptor. I n l i q u i d s , X is usually t y p i c a l o f rotational relaxation t i m e s , w h i c h , for systems w i t h large d i p o l e m o m e n t s , can give values o f \/fi as large as 1 0 s" . Solids p r o v i d e the a b i l i t y to c o n t r o l X as, i n p r i n c i p l e , geometries can b e c o n s t r u c t e d to favor some relaxation mechanisms b u t disfavor others. D r a m a t i c effects of this nature have b e e n o b s e r v e d i n the spectroscopy o f m i x e d crystals. I f t h e y c o u l d b e c o n t r o l l e d , t h e n it is possible that a m o l e c u l a r s w i t c h c o u l d operate b y m o d u l a t i n g the relaxation t i m e kh' i n o r d e r to change the d o n o r - t o acceptor c u r r e n t . 1 2

1

1

E l e c t r o n transfer adds a n extra d e g r e e o f c o m p l e x i t y to the basic e n e r g y transfer p r o b l e m i n that t w o different sets o f q u a n t u m n u m b e r s , e l e c t r o n i c a n d v i b r a t i o n a l , are i n v o l v e d . M u c h o f the c u r r e n t research o n e l e c t r o n transfer processes r e p o r t e d i n the c h e m i c a l l i t e r a t u r e is c o n c e r n e d w i t h the special cases i n w h i c h the v i b r a t i o n a l coordinates n e e d to b e treated e x p l i c itly. S u c h studies (e.g., refs. 46 a n d 47 a n d references therein) i n v o l v e b o t h v i b r a t i o n a l a n d electronic coordinates; t h e y are often c o n c e r n e d w i t h d e p e n d e n c e of the relaxation t i m e X o n the solvent f r i c t i o n . T h i s emphasis is not necessary, h o w e v e r . I n this chapter w e restrict ourselves to p r o b l e m s describable b y j u s t the electronic q u a n t u m n u m b e r s . A m o r e general t h e o r y w i l l b e p r e s e n t e d elsewhere (52).

General Formalism Let a n d |i|i ) b e wave functions r e p r e s e n t i n g the appropriate p h y s i c a l q u a n t i t y (here, t y p i c a l l y a n e l e c t r o n or h o l e function) r e p r e s e n t i n g the d o n o r a n d acceptor states, respectively. L e t the n wave functions \ty ) for a l l i = 1 - n represent similar quantities r e p r e s e n t i n g b r i d g e states. A l l of these functions are r e q u i r e d to b e orthogonal. T h e total H a m i l t o n i a n operator H for the system p l u s bath is expressed as A

t

H

= H

s

-

iX|ifo> = 2

C |x,>

(3)

#

As H is not h e r m i t i a n , e w i l l be c o m p l e x a n d the matrix of the eigenvectors Xjy may be b o t h not unitary a n d singular. Singularity is a rare p h e n o m e n o n , and its effects (or, rather, lack of effects) are discussed elsewhere (53). W h e n they are l i n e a r l y i n d e p e n d e n t , this i n v e r t i b l e transformation yields ;

2

l*> =

(4)

W i t h the assumption that the system is i n the d o n o r state |i(i ) at t i m e t = 0, t h e n the p r o b a b i l i t y P^t) of b e i n g i n state i at t i m e t is D

p,(t) = K » k k

i H

"1»M

=

2

S ^ V e x p - p - '

(5)

and the p r o b a b i l i t y P(t) that the excitation has not leaked to the bath is g i v e n by

P(f)

=

2

PM

=

2

Pa

exp

=

P*

=

2H(| *

P (t), a n d the p r o b a b i l i t y o f not d e c a y i n g to the b a t h , P(t) from e q 6, are p l o t t e d against r e d u c e d t i m e for the resonant p r o b l e m T| = 0. H e r e , w e take a = | H | . F i g u r e 2 is a p l o t o f e q 15 versus the r e d u c e d decay rate X / a . T h e rate constant exhibits t h r e e r e g i m e s : the slow d r a i n r e g i m e , i n w h i c h the rate constant increases i n p r o p o r t i o n to X; a m a x i m u m r e g i m e i d e n t i f i e d w i t h the m a x i m u m rate constant p r e d i c t e d b y R a b i ; a n d a r e g i m e i n w h i c h the rate constant slows w i t h i n c r e a s i n g b a t h decay rate. T h i s t h i r d r e g i m e is i d e n t i f i e d as a m i c r o s c o p i c f o r m of F e r m i ' s g o l d e n r u l e . T h e time-response functions g i v e n i n F i g u r e 1 are s a m p l e d from each of these three regimes. Rabi's m a x i m u m r e g i m e is the one that most closely couples the entrance a n d exit rates t h r o u g h the acceptor state, a n d i n fact o n l y a s m a l l amount of a m p l i t u d e b u i l d s u p o n this state. A

D A

A general analytical solution of e q 10 is difficult to o b t a i n , a n d so w e p r o c e e d b y e v o k i n g p e r t u r b a t i o n theory to d e t e r m i n e approximate expres sions for Hk i n the l i m i t s of s m a l l X a n d large X. T h e n w e c o m b i n e these results a n d construct b y i n t e r p o l a t i o n an expression appropriate for the e n t i r e range of X. Small-X Expansion. tonian H as

W h e n X is s m a l l , w e express the total H a m i l

H = H + AH, s

AH = -iXliM^I


(16)

34


T—I—i—|—i—|—i—|—i—|—i—i—i—|—r N

in


o

-

\R _L_

1

2

3

1

2

4

3

5

4 ttf/ffn

6

7

5

8

6

Figure 1. Symmetric resonant two-level system: plots of the probability of being in the acceptor state, P , and the probability of not decaying to the bath, P, versus reduced time ta/(irfi) for various acceptor-to-bath coupling strengths X. Key: N, no decay, X = 0; S, slow drain regime, X = O.J X 2 AOL; R, Rabi maximum rate, X = 2 Aa; and G, golden rule regime, X = 10 X 2 /ia (this curve barely exceeds zero). A

1

l

l

a n d d e t e r m i n e the u n i t a r y eigenvector matrix I T a n d the associated diagonal eigenvalue m a t r i x E ' of the system H a m i l t o n i a n H . T e r m s c o n t a i n i n g X are s

treated as perturbations a n d are exposed b y t r a n s f o r m i n g H to p r o d u c e H' = (u') mr f

= E' -

AH'

(17)

where AH/

=

- i x i v i y


(18)

2.

35

Bridged Systems

REIMERS & H U S H

i — r - — T — i — | — i — i — i — r - —1—i—i—i—i—|—i—i—l—l—|—i—i—i—l—|—i

1—l—i—

R A B I MAXIMUM fik=«/y2


o

£ ^. 1 i

SLOW D R A I N

o Ol

at

X=V2a

/

hk=X

\

GOLDEN

RULE!

\fil)

+

GR

12

2

(

j

w h e r e n / a n d n ^_are defined from the small-X l i m i t b y eqs 22 a n d 23, respectively, a n d V is defined from the golden r u l e l i m i t b y e q 36. T h i s function leads to a definition of an approximate rate constant k


a

1 2

( G R )

i9

2X(V < >)

_

=

'

GR

12

2

(V < >) n V + X 12

GR

2

a

2

(

0

)

w h i c h does i n fact have the correct l i m i t s as X —• 0 a n d as X —> a

r

GR

12

(46)

a n d occurs at X = X , w h e r e i R

x< = KV) V < > R

V2

12

GR

(47)

so that

Am = K'nMiK

(48)

and ttmX»

= (V < ») 12

GB

2

(49)

W h e n e q 46 is a p p l i e d to t w o - l e v e l systems, e q 15 is o b t a i n e d (even for arbitrary i\) so that this f o r m u l a i n fact gives exact results. E m p i r i c a l l y , i t is o b s e r v e d that e q 46 is always exact to w i t h i n n u m e r i c a l p r e c i s i o n w h e n f u l l q u a n t u m y i e l d is obtained. T h u s , the e n t i r e X d e p e n d e n c e o f the rate c o n stant may be expressed e x p l i c i t l y . A l l of the other quantities i n e q 46 can b e o b t a i n e d from a k n o w l e d g e o f just the isolated system H a m i l t o n i a n , p r o v i d i n g a considerable p h i l o s o p h i c a l a n d c o m p u t a t i o n a l advance. T h i s g e n eral result is interesting, b u t no formal p r o o f o f this equivalence is yet k n o w n . F r o m a practical standpoint, the m e t h o d of extrapolation is s i m p l e r to i m p l e m e n t than the i n t e r p o l a t i o n m e t h o d because e q 46 contains parameters


2.

Bridged Systems

REIMERS & H U S H

41

d e r i v e d from t w o different p e r t u r b a t i o n expansions u s i n g two different ref e r e n c e H a m i l t o n i a n s , eqs 16 a n d 26. T h e advantage of u s i n g k is, h o w e v e r , A

that it is m o r e accurate. F o r almost a l l p r o b l e m s of p r a c t i c a l i m p o r t a n c e , fc ' R

is sufficiently accurate, a n d thus this extra accuracy is i m m a t e r i a l . I f

accuracy forces one to use k

i9

t h e n , i n c e r t a i n i m p o r t a n t l i m i t s , the e i g e n

states of one o f the two H a m i l t o n i a n s can b e expressed i n t e r m s of the


eigenstates of the other, thus greatly s i m p l i f y i n g the theory.

Relation to the Work ofJoachim T h e R a b i m a x i m u m r e g i m e is v e r y useful i n that it reveals the m a x i m u m rate constant, i r r e s p e c t i v e o f the nature of the e n v i r o n m e n t . Its analogous r e g i m e for p r o b l e m s i n w h i c h b o t h electronic a n d v i b r a t i o n a l coordinates m u s t b e c o n s i d e r e d is treated successfully b y Beats's g e n e r a l i z e d m e t h o d (29). I n a w e l l - d e s i g n e d system, the " o n " state of some m o l e c u l a r s w i t c h w i l l have a transfer rate close to this m a x i m u m rate, a n d so this r e g i m e is i m portant to the design of m o l e c u l a r electronic devices. J o a c h i m (39) d e v e l o p e d a theory for this r e g i m e i n b r i d g e d systems that has p r o v i d e d m u c h insight i n t o the causes of d i s t a n c e - i n d e p e n d e n t transfer processes. H i s theory is i n s p i r e d , b u t a d hoc, i n that no p h y s i c a l basis exists for the i m p l i c i t postulates. A s a result, n o n p h y s i c a l effects s u c h as d i s c o n tinuities i n the m a x i m u m rate constant as a f u n c t i o n o f the H a m i l t o n i a n are i n t r o d u c e d . T h i s chapter demonstrates that, u n d e r c e r t a i n c o n d i t i o n s , o u r general f o r m u l a for the R a b i rate constant reduces to a f o r m v e r y s i m i l a r to the equations of J o a c h i m , b u t c o n t a i n i n g n o n e of the n o n p h y s i c a l effects. W e w o r k e d from the extrapolated f o r m u l a , e q 39, a n d assumed that no b r i d g e states are resonant w i t h e i t h e r the d o n o r o r acceptor states. 0 o a c h i m ' s t h e o r y is i n t e n d e d to treat resonant situations.) N o w o n l y one off-diagonal p - m a t r i x e l e m e n t (see e q 7) remains significant, a n d this contributes

(e ' d

-

(50)

Ey A

to b (see e q 21), w h e r e D a n d A are the indices of the eigenstates w h o s e character is d o m i n a t e d b y the donor a n d aeeeptorbasis states, r e s p e c t i v e l y . T h e effective c o u p l i n g e l e m e n t for the R a b i rate, V (see e q 42), simplifies i n this nonresonant l i m i t to 1 2

T7 (NR) _ V l 2

"

1 P' £

£

A

( R )

1

(2n 'n y* a

r

/Kl\ (

}

If w e consider o n l y p r o b l e m s i n w h i c h the d o n o r a n d acceptor are r e l a t e d b y s y m m e t r y , t h e n n ' = 1. A l s o , i n these circumstances, i t is s h o w n else w h e r e (53) that w h e n the q u a n t u m y i e l d is h i g h b u t < 1 (say 0.99), t h e n the a


42


n u m b e r o f resonant modes n / - » 2. E q u a t i o n 39 thus simplifies to

M

S N R

(52)

' = 2-'' V < > 2

SNB

12

where


(SNR)

J o a c h i m (39) p r o p o s e d t w o expressions for t h e R a b i rate constant a n d c a l l e d them the " H

e f f

" m e t h o d a n d t h e " E x a c t " m e t h o d . T h e exact m e t h o d i n v o l v e d

the solution o f t h e t i m e - d e p e n d e n t Hamiltonian H , a n d the H s

S c h r o d i n g e r e q u a t i o n for t h e system

m e t h o d is d e r i v e d as a n a p p r o x i m a t i o n t o t h e

e f f

exact m e t h o d . R a t h e r , w e demonstrate that h i s H

m e t h o d is closer to

e f f

p h y s i c a l reality a n d that h i s exact m e t h o d u s u a l l y p r o v i d e s o n l y a r o u g h n u m e r i c a l a p p r o x i m a t i o n to i t . J o a c h i m a p p l i e d h i s m e t h o d s to systems d e s c r i b e d b y matrices o f the f o r m suggested b y M c C o n n e l l (24) P 0 : 0 0 0\

/0

p e

0

a l 0 0 0

0 a e

0

: 0 0 0 (53)

0

0 0 : e

0

0 0 : a e

0

a 0 0

p

\0 0 0 : 0 p 0 / where H

DA

H

ii+1

= 0, H

DD

= H^, H

BB

= H

M

+ e, H 0

D1

= p, and

= H

An

= a for a l l i a n d i + 1 o n t h e b r i d g e . T h e exact m e t h o d is based o n a s i m p l e extension o f R a b f s o r i g i n a l

concept. Rabi's i d e a , appropriate for t w o - l e v e l systems, is that t h e m a x i m u m possible rate is t h e inverse o f t h e t i m e . I n t h e no-reaction case, w h e r e a m p l i t u d e just pendulates from d o n o r to acceptor a n d back again, i t takes the o r i g i n a l a m p l i t u d e i n t h e d o n o r to first r e a c h t h e acceptor. T h i s process is i l l u s t r a t e d i n F i g u r e 3 w h e r e , for a t w o - l e v e l system, t h e p r o b a b i l i t y o f b e i n g i n t h e acceptor state P is p l o t t e d against r e d u c e d time tV l(ITft). A

12

It is easy to show that t h e p e n d u l a t i o n t i m e o f t h e first donor-to-acceptor

frequency

is o> = 2V /ti, 12

so t h e

t r a n s i t i o n is IT/A,D

AA

rt

ZJ j*A,D

QT *"

\

h

H

p

F

w

^

T? ^

(68) "

T? W. &j

after one cycle of L o w d i n p a r t i t i o n i n g (55, 56).


50


U n f o r t u n a t e l y , the parameter E i n this e q u a t i o n is i l l - d e f i n e d , a n d this construction yields a u n i q u e result o n l y for the v e r y i m p o r t a n t case w h e n H = H . I n general, B r o o a n d L a r s s o n (45) expressed this p a r a m e t e r i n an ad hoc fashion as the average E = (H + O t h e r attempts (57) DD

M

DD

to resolve this a m b i g u i t y generate models w i t h other adverse p r o p e r t i e s such as n o n h e r m i t i a n system-only H a m i l t o n i a n s H . T y p i c a l l y , E is m u c h


s

larger than the energy difference H ^ - H p o , a n d so the precise d e f i n i t i o n o f E is not of great practical significance. O u r approach p r o v i d e s a u n i q u e resolution of this a m b i g u i t y , b u t e q 64 also contains a n adverse p h y s i c a l p r o p e r t y . T h e microscopic forward a n d reverse rate constants calculated b y u s i n g e q 64 are different because o f the u n i q u e role g i v e n to H i n e q 65. B r o o a n d Larsson's (45) resolution of the a m b i g u i t y i n E does, c o r r e c t l y , show microscopic r e v e r s i b i l i t y . I n o u r results, the a s y m m e t r y results from the v e r y different p h y s i c a l roles ascribed to the donor a n d acceptor states. I f necessary, an i m p r o v e d rate constant c o u l d b e d e f i n e d as the r e c i p r o c a l of h a l f of the t i m e r e q u i r e d to c o m p l e t e m i c r o s c o p i c a l l y b o t h the f o r w a r d a n d reverse reactions, k = 2k k /(k + fc ). DD

F

R

F

R

Rabi Maximum. N e x t , w e investigated the R a b i m a x i m u m r e g i m e . I n Larsson's approach (43-45), one effective c o u p l i n g e l e m e n t is generated a n d u s e d i n a t w o - l e v e l kinetics theory to d e t e r m i n e the reaction-rate c o n stant. W e p r o c e e d e d b y e x p a n d i n g o u r general f o r m u l a for the R a b i rate constant, e q 52, i n the l i m i t of a nonresonant b r i d g e . F o r p r o b l e m s i n w h i c h the d o n o r a n d acceptor states are r e l a t e d b y s y m m e t r y , o u r approach a n d the approach of B r o o a n d L a r s s o n (45) give the same results. W h e n T| is large, application of f o u r t h - o r d e r p e r t u r b a t i o n theory y i e l d s

E' D

= H

-

DD

tf> (H

M

+

2

1

? "

j * A , D "DD

\

H

~

i ) ' &j

(69)

I

and

E' A

=

H

A

A

-

V

(H

DA

\

+

2

TT^T

j#A,D " A A

~~

I

™

w h e r e E", H ", a n d H " are g i v e n i n eqs 58 a n d 6 1 . T h e s e equations preserve microscopic r e v e r s i b i l i t y , a n d t h e y cannot b e w r i t t e n i n the f o r m g i v e n b y an effective t w o - l e v e l theory. H o w e v e r , as E is m u c h larger t h a n its u n c e r t a i n t y , v e r y s i m i l a r results are expected from the t w o approaches. D

Aj

I n practical c o m p u t a t i o n a l t e r m s , no great difference exists b e t w e e n the m e t h o d of B r o o a n d L a r s s o n (43-45) a n d the methods p r e s e n t e d h e r e . B o t h sets of expressions appear expensive to evaluate for large b r i d g e s , as t h e y r e q u i r e the k n o w l e d g e of a l l of the eigenvalues a n d eigenvectors of the system


2.

REIMERS & H U S H

Bridged Systems

51

H a m i l t o n i a n . H o w e v e r , the equations can be r e w r i t t e n (53) i n a f o r m that can b e solved i t e r a t i v e l y w i t h o u t r e q u i r i n g any eigenvalues to b e f o u n d or e v e n the entire matrix to be simultaneously h e l d i n core m e m o r y . T h i s approach makes practical calculations i n v o l v i n g m a n y thousands of b r i d g e states. T h e concepts u n d e r l y i n g Larsson's approach have b e e n a r o u n d for a l o n g t i m e a n d have p r o v e d v e r y successful i n i n t e r p r e t i n g nonresonant e l e c tron-transfer processes. B r o o a n d L a r s s o n (45) c e m e n t e d these ideas i n t o a Downloaded by UNIV OF NEW SOUTH WALES on September 3, 2015 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1990-0226.ch002

strong s i m p l e u n i t . F r o m the foregoing discussion, it is reasonable to c o n c l u d e that o u r approach reduces to B r o o a n d Larsson's i n the nonresonant l i m i t , as i t s h o u l d .

Relation to the Work of McConnell M a n y years ago, M c C o n n e l l (24) p r e s e n t e d a v e r y useful f o r m u l a for i n t e r p r e t i n g s p i n transfer processes i n situations w h e r e the b r i d g e levels are nonresonant w i t h the d o n o r a n d acceptor levels. I n a w e l l - d e s i g n e d system, the "off* state of some m o l e c u l a r electronics d e v i c e w i l l most l i k e l y display kinetics i n this r e g i m e (5,6); thus, it is i m p o r t a n t to u n d e r s t a n d the properties of this r e g i m e . Today, c h e m i c a l physics interest is often d i r e c t e d t o w a r d p r o b l e m s i n w h i c h resonance occurs, a n d for these p r o b l e m s M c C o n n e l l ' s approach is i n a p p l i c a b l e . A u t h o r s often c o m p a r e the results of t h e i r advanced theories for resonant rate constants to the results o f M c C o n n e l l ' s theory w i t h o u t discussion. F r o m r e a d i n g , one is left w i t h the f e e l i n g that M c C o n n e l l ' s f o r m u l a s h o u l d be d i s c a r d e d a n d o n l y the n e w f o r m u l a r e t a i n e d . I n ref. 39, a factor of 2 has adventitiously e n t e r e d i n t o M c C o n n e l l ' s f o r m u l a , a n d so J o a c h i m failed to notice that his f o r m u l a does i n fact give the correct n o n resonance results. I n this chapter, w e r e i n t e r p r e t M c C o n n e l l ' s w o r k i n the l i g h t o f m o d e r n discussion a n d show h o w his s i m p l e analytical results can b e i n c l u d e d at the appropriate l i m i t of m o r e - g e n e r a l theories. T h e usefulness o f M c C o n n e l l ' s f o r m u l a stems f r o m the practical diffi culties i n evaluating equations l i k e eqs 5 1 , 65, a n d 69 for large b r i d g e s . A n efficient iterative m e t h o d for evaluating these equations is d e s c r i b e d else w h e r e (53). M c C o n n e l l ' s f o r m u l a provides an analytical solution for the rate constant of systems d e s c r i b e d b y n + 2 d i m e n s i o n a l H a m i l t o n i a n matrices of the f o r m o f e q 53. T h i s m o d e l H a m i l t o n i a n is excellent for s t u d y i n g t h r o u g h - b r i d g e k i n e t i c s . M c C o n n e l l (24) s h o w e d that, for s u c h systems,

T h i s f o r m u l a is e x t r e m e l y useful, as it d i r e c t l y relates m a t r i x e l e m e n t s to rate constants. C l e a r l y , it is appropriate o n l y i n the nonresonant b r i d g e l i m i t ,


52


and methods such as those of J o a c h i m (39) a n d of B r o o a n d L a r s s o n (45) r e d u c e to M c C o n n e l F s f o r m u l a i n this l i m i t .


Model of Resonant Bridge Coupling E q u a t i o n 46 is o b s e r v e d n u m e r i c a l l y to p r o v i d e exact results for the rate constant w h e n f u l l q u a n t u m y i e l d is o b t a i n e d . It is m u c h s i m p l e r than the general expression, a n d it m a y b e a p p l i e d to o b t a i n analytical expressions for the rate constant of c o m p l i c a t e d systems. I n d e s i g n i n g a m o l e c u l a r elec tronics device to have particular d e s i r e d p r o p e r t i e s , one has a great advantage i f an analytical expression is available for the c u r r e n t (rate constant) as a function o f the m o l e c u l a r parameters. W e adopted a t h r e e - l e v e l system as the basic m o d e l for electron transfer t h r o u g h a resonant b r i d g e . T h e m e t h o d a p p l i e d here may be a p p l i e d to generate analytical models for m o r e c o m p l i c a t e d systems, i f d e s i r e d . W e neglected the effects o f through-space c o u p l i n g , a n d w r o t e the t h r e e - l e v e l H a m i l t o n i a n as IH

0

H

DD

H

DB

= JH

DB

H

0

H

BA

ikj

HAA -

BA

(72)

I

H

BB

F o r s i m p l i c i t y , w e e l i m i n a t e d the zero of energy b y i n t r o d u c i n g the variables e

= H

0

B

B

- H

D

D

a n d TJ = H

A

- H

A

D

D

. T h i s step leaves five i n d e p e n d e n t

variables. To evaluate e q 46, w e first o b t a i n e d V

1 2

r G R )

b y u s i n g e q 36. T h i s c a l

culation gives V

(GB)

HDBHBA

=

(2ff

(

+ e

Formalism for Electron Transfer and Energy Transfer in Bridged Systems

Recommend Documents