Electron Transfer in Inorganic, Organic, and Biological Systems

5 Electron Transfer

Downloaded by UNIV OF ARIZONA on November 17, 2012 | http://pubs.acs.org Publication Date: May 5, 1991 | doi: 10.1021/ba-1991-0228.ch005

From Model Compounds to Proteins David N. Beratan and José Nelson Onuchic 1

2,3

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 Department of Physics, University of California at San Diego, La Jolla, C A 92093 and Institute de Fisica e Quimica de São Carlos, Universidade de São Paulo, São Carlos, SP, Brazil

1

2

We summarize the formulation of the protein-mediated electronic coupling calculation as a two-level system with weakly interacting bridge units. Using model compounds as a starting point from which to derive coupling parameters, we present a strategy for defining the pathways for electron tunneling in biological andbiomimeticalsystems. The specific bonding and nonbonding interactions in cytochrome c and myoglobin that mediate the tunneling between the porphyrin and an attached transition metal probe are described. The method appears to succeed where traditional structureless tunneling barrier or periodic bridge models are not adequate. An algorithm to search for these tunneling pathways in proteins is described, and the nature of the paths is discussed.

THE PROCESS OF ELECTRON TRANSPORT IS CENTRAL

i n chemistry, biology, a n d physics. T h i s field is f r e q u e n t l y subjected to d e t a i l e d reanalysis a n d r e v i e w (1-4). W e b e g i n the discussion h e r e b y p r e s e n t i n g the H a m i l t o n i a n that has b e e n u s e d extensively to m o d e l the generic electron-transfer p r o b lem

H

3

E T

= H^

x

+ | δσ

ζ

+ H

Q

(1)

Current address: Department of Physics, University of California at San Diego, La Jolla, CA 92093 0065-2393/91/0228-0071$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

Η is t h e t u n n e l i n g m a t r i x e l e m e n t b e t w e e n d o n o r a n d acceptor (reactants a n d products); σ a n d σ are t h e P a u l i matrices, w h e r e t h e e i g e n v a l u e σ = 1 is associated w i t h t h e reactant state a n d the eigenvalue σ = - 1 is associated w i t h t h e p r o d u c t state; H supplies t h e d y n a m i c s for t h e n u c l e a r m o t i o n (reaction coordinates a n d bath), a n d δ is t h e instantaneous e n e r g y difference b e t w e e n t h e reactants a n d products (5-8). T h i s H a m i l t o n i a n leads to t h e u b i q u i t o u s rate e q u a t i o n for transfer b e t w e e n w e a k l y c o u p l e d donors a n d acceptors (1-10). φ

χ

ζ

ζ

ζ

Q


H J

2

(FC)

(2)

A s s u m i n g that this separation can b e p e r f o r m e d a n d that t h e process is n o t relaxation-controlled, t h e rate is p r o p o r t i o n a l to the e l e c t r o n i c c o u p l i n g factor |Η | times a n u c l e a r F r a n c k - C o n d o n ( F C ) w e i g h t e d d e n s i t y o f states (ac tivated) factor; h is Planck's constant d i v i d e d b y 2ττ. E q u a t i o n 2 gives t h e rate i n the weak c o u p l i n g l i m i t , often c a l l e d t h e nonadiabatic l i m i t . T w o i m p o r t a n t conditions must h o l d to w r i t e this e q u a t i o n . F i r s t , a n energy separation is r e q u i r e d to r e d u c e t h e p r o b l e m to t h e H a m i l t o n i a n g i v e n b y e q 1 (i. e., a t w o - l e v e l system c o u p l e d to n u c l e a r m o d e s ; r e n o r m a l i z a t i o n procedure). A separation of electronic energies is also r e q u i r e d so that t h e electronic p r o b l e m c a n b e r e d u c e d to a t w o - l e v e l (donor a n d acceptor) system. S e c o n d , e v e n w h e n t h e H a m i l t o n i a n of e q 1 is v a l i d , the transfer m u s t b e nonadiabatic to w r i t e the rate i n e q 2 (1-10). T h e nonadiabatic l i m i t is v a l i d for t h e m o d e l systems a n d proteins discussed i n this chapter. N e x t w e discuss w h y the s i m p l e H a m i l t o n i a n (eq 1) is a p p r o priate for such c o m p l e x p r o b l e m s . φ

2

Bridge-Mediated Electron Tunneling and Two-Level Systems F i r s t , consider t h e electronic part of this p r o b l e m . Because of the c o m p l e x i t y of p r o t e i n s , w e h o p e to r e d u c e i t to s m a l l e r appropriate parts (if possible) that c a n b e a n a l y z e d a n d u n d e r s t o o d . T h i s is a c h i e v e d b y gradually e l i m i n a t i n g h i g h e r energies. T h e first step i n this p r o c e d u r e is to assume that the energies i n v o l v e d i n c h e m i c a l b o n d i n g are v e r y s m a l l c o m p a r e d to c o r e e l e c t r o n excitations. T h i s a s s u m p t i o n allows t h e e l i m i n a t i o n o f the core e l e c trons. T h e core electrons p r o v i d e a p s e u d o p o t e n t i a l i n w h i c h t h e valence electrons m o v e . N e x t w e m a k e t h e f u r t h e r a s s u m p t i o n that c o u p l i n g energies associated w i t h h o p p i n g b e t w e e n n e i g h b o r i n g b o n d i n g orbitals are s m a l l c o m p a r e d to atomic excitation energies. T h i s assumption leads to a t i g h t b i n d i n g m o l e c u l a r o r b i t a l p i c t u r e (J J). T h e s e assumptions are generally v a l i d for t h e electron-transfer p r o b l e m a n d are a d o p t e d t h r o u g h o u t this chapter (4, 5). T h i s m o d i f i c a t i o n justifies an effective o n e - e l e c t r o n H a m i l t o n i a n a n d p e r m i t s c o m p u t a t i o n o f the t u n -


5.

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73

From Model Compounds to Proteins

n e l i n g m a t r i x e l e m e n t Η . F i n a l l y , to r e d u c e the o n e - e l e c t r o n H a m i l t o n i a n of the e n t i r e system to a t w o - l e v e l H a m i l t o n i a n , t h e e l e c t r o n i c e n e r g y sep aration b e t w e e n d o n o r (or acceptor) a n d b r i d g e sites m u s t b e m u c h l a r g e r than the c o u p l i n g energy b e t w e e n d o n o r a n d acceptor. I f this is not the case, there are e l e c t r o n i c excitations w i t h energies o f the same o r d e r as t h e d o n o r - a c c e p t o r c o u p l i n g , i n v a l i d a t i n g a t w o - l e v e l approach. [These e n e r g y comparisons are best m a d e w i t h b o n d orbitals r a t h e r t h a n a t o m i c orbitals


φ

(12).] W e n o w i n c l u d e t h e v i b r a t i o n a l modes. I f the energy scales associated w i t h excitations o f these modes are m u c h smaller t h a n t h e e l e c t r o n i c e x c i tation energies, w e c a n use t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n . T h i s a p p r o x i m a t i o n allows us to solve t h e electronic p r o b l e m for fixed n u c l e a r coordinates, so t h e n u c l e a r coordinates e n t e r as parameters. A t w o - l e v e l system t h e n results, w i t h energies that are functions o f n u c l e a r coordinates. T h e t u n n e l i n g m a t r i x e l e m e n t is calculated b y fixing the n u c l e a r coordinates so that t h e reactant a n d p r o d u c t states have t h e same e n e r g y ( C o n d o n a p proximation) (2, 4, 5). I f a l l o f t h e energy separations discussed h e r e are a p p r o p r i a t e , t h e p r o b l e m is r e d u c e d to e q 1. References 5 a n d 13 d e s c r i b e details o f the e l e c t r o n i c - n u c l e a r energy separation. A tutorial s h o w i n g t h e r e d u c t i o n o f a t h r e e - l e v e l system to a t w o - l e v e l system is g i v e n i n ref. 5. E l e c t r o n i c excitation energies are about 1 - 3 eV, so this separation is v a l i d for most o f the n u c l e a r modes. N e x t , w e i n c l u d e t h e h i g h - f r e q u e n c y n u c l e a r m o d e s (C = Ο stretches, for e x a m p l e , w i t h i n t h e range 0 . 1 - 0 . 2 5 e V ; Ω is t h e f r e q u e n c y o f the mode). I n this case, h£l is m u c h larger t h a n other v i b r a t i o n a l excitation energies a n d k T (k is B o l t z m a n n ' s constant a n d Γ is t h e temperature). T h e s e m o d e s , w h i c h t y p i c a l l y arise from local v i b r a t i o n s , have a n e a r l y B

B

discrete s p e c t r u m [very l o w d a m p i n g (14)], w h i c h s h o u l d b e t r e a t e d i n t h e quantum limit; they simply renormalize the tunneling matrix element and the d r i v i n g force (7). F o r e x a m p l e , i n a t w o - l e v e l system w i t h t h e r m o d y n a m i c d r i v i n g force δ c o u p l e d to one h i g h - f r e q u e n c y m o d e , | n > (|n >) represents the v i b r a t i o n a l state o f the h i g h - f r e q u e n c y m o d e w h e n t h e e l e c t r o n is o n the d o n o r (D) o r acceptor (A). (The e q u i l i b r i u m p o s i t i o n o f this h i g h - f r e q u e n c y m o d e shifts, d e p e n d i n g o n w h e t h e r t h e e l e c t r o n is o n t h e d o n o r o r acceptor.) Because k T « M l , t h e d o n o r v i b r a t i o n a l state is always | 0 > . O n e o f the acceptor states | n > w i l l d o m i n a t e t h e process, d e p e n d i n g o n δ . T h e r e n o r m a l i z e d parameters are 0

D

A

B

D

A

0

H^

=

s

Η

δο * = δ 6

0

φ

(3a)

n Ml

(3b)

ο

-

A

T h e effective d o n o r state is | D > | 0 > , a n d t h e effective acceptor state is | A > | n > (el signifies a n e l e c t r o n i c state). F i n a l l y , i f the e l e c t r o n i c e x c i t a el

el

D

A


74



tions are of the same o r d e r as hit a n d the reorganization energy is a few times M l , the r e n o r m a l i z a t i o n p r o c e d u r e is a b i t different a n d w e m u s t construct an energy cutoff that includes the electronic states a n d the h i g h f r e q u e n c y m o d e . W e cannot separate this process into two stages as w e d i d before (electronic part first, t h e n the h i g h - f r e q u e n c y mode). T h e final result is v e r y s i m i l a r to the present one (i.e., a single d o n o r state a n d a set of discrete acceptor states), b u t these states w i l l be m i x t u r e s of e l e c t r o n i c a n d h i g h - f r e q u e n c y n u c l e a r states rather than s i m p l e products. E n e r g y separation is not the o n l y r e q u i r e m e n t for the v a l i d i t y of the B o r n - O p p e n h e i m e r a p p r o x i m a t i o n . A l t h o u g h energy separation guarantees that w e can neglect the d o n o r (or acceptor) excited e l e c t r o n i c states, care is r e q u i r e d w h e n c o m p u t i n g the t u n n e l i n g matrix e l e m e n t that d e p e n d s o n details of the electronic wave function tail. F o r m a l l y , as the e l e c t r o n m o v e s f r o m d o n o r to acceptor it spends an i m a g i n a r y t i m e (a traversal time) i n the f o r b i d d e n r e g i o n (15). I f the n u c l e a r modes are slow c o m p a r e d to this t i m e , the B o r n - O p p e n h e i m e r a p p r o x i m a t i o n works (i.e., the n u c l e i stay essentially fixed as the e l e c t r o n tunnels). T h e traversal t i m e increases w i t h the t u n n e l i n g distance a n d decreases w i t h the t u n n e l i n g b a r r i e r h e i g h t (5). F o r v e r y - l o n g range transfer the B o r n - O p p e n h e i m e r a p p r o x i m a t i o n m u s t break d o w n , b u t this a p p r o x i m a t i o n is reasonable for the systems discussed h e r e (4, 16). T o this p o i n t , w e have d e s c r i b e d w h y the H a m i l t o n i a n i n e q 1 is, i n m a n y cases, an appropriate starting p o i n t for the electron-transfer p r o b l e m . W e w i l l n o w describe h o w to obtain the t w o - l e v e l representation of the e l e c t r o n i c p o r t i o n of the p r o b l e m for b r i d g e d systems. T h e questions to b e addressed are (1) W h a t are these two states i n a c o m p l e x b r i d g e d system? (2) H o w is the c o u p l i n g Η b e t w e e n the two states related to energy splittings that are o b t a i n e d from electronic structure calculations? T h e simplest example of b r i d g e - m e d i a t e d electron transfer i n a t i g h t b i n d i n g or m o l e c u l a r o r b i t a l m o d e l results i n the H a m i l t o n i a n of e q 4. T h e d o n o r a n d acceptor are o n l y c o u p l e d b y t h e i r m u t u a l interactions w i t h one b r i d g e (B) o r b i t a l v i a the exchange interactions β and β . The Hamiltonian matrix i n this case ( a > a , a ) is φ

Ο Β

B

D

Β Α

A

(4)

T h e n u c l e a r coordinates are r e p r e s e n t e d b y y, a n d e q 4 is w r i t t e n i n the B o r n - O p p e n h e i m e r a p p r o x i m a t i o n . T h e y d e p e n d e n c e of the site energies reflects the separation b e t w e e n electronic a n d n u c l e a r m o t i o n a n d the as s u m p t i o n that o n l y the d o n o r a n d acceptor orbitals are c o u p l e d to n u c l e a r distortions. Because the donor a n d acceptor (unmixed) orbitals are d e g e n erate at the crossing of the n u c l e a r surfaces, a (y) = a (y) = α (Condon O

A


5.

BERATAN & ONUCHIC


75

approximation). T h e s y m m e t r i e - a n t i s y m m e t r i e s p l i t t i n g (ΔΕ) b e t w e e n t h e two lowest states l o c a l i z e d d o m i n a n t l y o n d o n o r a n d acceptor (eq 4) is

(α

Β

-

α) ^ ( β

2

+ β

(α„ -

2


Ρ Β

Β Α

2

)

(5)

α)

T h i s s p l i t t i n g is n o n z e r o e v e n w h e n there is no d o n o r - b r i d g e o r b r i d g e acceptor c o u p l i n g ! C o n t r a r y to the c o m m o n c l a i m , this s p l i t t i n g is n o t p r o p o r t i o n a l to Η . T h e r e s o l u t i o n of this issue arises f r o m t h e fact that w e have calculated t h e splitting b e t w e e n the w r o n g states. T h e s p l i t t i n g i n e q 3 is n o n z e r o because i t i n c l u d e s contributions o f p u r e d o n o r - b r i d g e a n d bridge-acceptor mixing. T h e net bridge-mediated donor-acceptor interac t i o n , Η , is n o t t h e s p l i t t i n g b e t w e e n states i n t h e o v e r a l l H a m i l t o n i a n w i t h a = a . A l s o , 21^/^1 is the energy associated w i t h m i x i n g t h e d o n o r plus b r i d g e state w i t h t h e acceptor plus b r i d g e state (i.e., t h e s p l i t t i n g o f t h e states i n t h e c o r r e s p o n d i n g t w o - l e v e l system). T h e splittings b e t w e e n e i genvalues of t h e f u l l H a m i l t o n i a n of e q 4 are not d i r e c t l y r e l a t e d to Η . F r o m t h e standpoint of p e r t u r b a t i o n theory, the d o n o r - a c c e p t o r degeneracy at t h e crossing p o i n t o f the n u c l e a r surfaces is b r o k e n o n l y i n second o r d e r b y t h e b r i d g e , so that the c o u p l i n g b e t w e e n the states i n this o r d e r is -β β /(α - α ) (17). O n l y i n a t r u e two-site m o d e l is t h e r e d i r e c t e q u i v alence b e t w e e n Δ Ε a n d 2|Η |. A l s o , strictly speaking, the o r b i t a l energies that w e r e m a d e e q u i v a l e n t i n t h e C o n d o n a p p r o x i m a t i o n s h o u l d b e t h e energies o f t h e t w o - l e v e l system, not t h e i n d i v i d u a l site energies o f t h e d o n o r a n d acceptor. φ

φ

D

A

ψ

Ο Β

Α Β

Β

φ

A general t e c h n i q u e to r e d u c e a b r i d g e d d o n o r - a c c e p t o r system to t h e c o r r e s p o n d i n g t w o - l e v e l system is Lôwdin diagonalization (18,19). W o r k i n g in a basis diagonal i n t h e b r i d g e orbitals, t h e total H a m i l t o n i a n is

/ « D

β 'DA :

βί>1

βί»Λ

β 1

β

Α

0 0

Η =

\ βίίϋ

β#Α

0

Α

Β

ο ο

0


(6)

76


P r i m e s d e n o t e interactions b e t w e e n a single atomic o r b i t a l a n d a m o l e c u l a r o r b i t a l a n d Ν is the n u m b e r of b r i d g e orbitals. U n p r i m e d interactions are b e t w e e n single a t o m i c orbitals. T h e exact c o r r e s p o n d i n g t w o - l e v e l H a m i l t o n i a n is

/ Η

=

i β


\

ϋ

Α

βίϋ ι

r

- Σ - Σ i

.«Bi

-

Ε

i

' βάβ* ' α

a

Bj

-

- Σ

Α

Ε_

Γ

- Σ

βϋΑ

i

βόίβΑί

_α„ -

Γ

1

Ε_

βΑΪ

.«» -

1 £.

T h e off-diagonal e l e m e n t s i n e q 7 are the e l e c t r o n t u n n e l i n g m a t r i x e l e m e n t s o f the c o r r e s p o n d i n g t w o - l e v e l system. T h e t u n n e l i n g energy Ε is d e t e r m i n e d b y the diagonal energies (these are d o n o r p l u s b r i d g e a n d acceptor plus b r i d g e energies) a n d the v i b r o n i c c o u p l i n g i n the m o l e c u l e (a s i m p l e average is a p p r o p r i a t e , for e x a m p l e , i f the v i b r o n i c c o u p l i n g o n the t w o sites is identical) (4). T h e r e are other methods of c a l c u l a t i n g t u n n e l i n g m a t r i x e l e m e n t s i n b r i d g e d systems. A n elegant m e t h o d that is e x p e r i e n c i n g g r o w i n g interest is the G r e e n ' s f u n c t i o n t e c h n i q u e . T h e m a t r i x elements o f the b r i d g e G r e e n ' s f u n c t i o n c o n t a i n the effective c o u p l i n g b e t w e e n sites i n the b r i d g e (20—22). N u m e r i c a l t e c h n i q u e s a p p l i c a b l e to G r e e n ' s functions are somewhat different from those u s u a l l y a p p l i e d i n a S c h r o d i n g e r e q u a t i o n a p p r o a c h , a n d some p o w e r f u l theorems a l l o w b o t h exact a n d p e r t u r b a t i o n evaluation of the c o u p l i n g s for t i g h t - b i n d i n g H a m i l t o n i a n s . T h e G r e e n ' s f u n c t i o n for a s y s t e m , G , is d e n n e d b y D y s o n ' s e q u a t i o n : (E -

= 1

H)G

(8)

I f the G r e e n ' s f u n c t i o n of the isolated b r i d g e is g i v e n b y G , the d o n o r is c o u p l e d to b r i d g e orbitals i w i t h strength β a n d the acceptor is c o u p l e d to sites η w i t h strength β . ϋ ί 5

ίΑ

H r p = β θ Α + ΣΣ i

βθΑηβ

ηΑ

(9)

η

G describes the propagation o f a m p l i t u d e w i t h i n the b r i d g e f r o m site i to site η; β is the d i r e c t " t h r o u g h - s p a c e " d o n o r - a c c e p t o r c o u p l i n g a n d can g e n e r a l l y b e n e g l e c t e d relative to the b r i d g e - m e d i a t e d terms for distant e l e c t r o n transfer. in

Ο Α

Information Learned from Model Compounds D o n o r p l u s b r i d g e a n d acceptor p l u s b r i d g e states are n e e d e d for a t w o l e v e l calculation o f H^. A s s u c h , techniques that calculate this m i x i n g r e l i a b l y


5.


77



w e r e the first targets o f study. A b i n i t i o t e c h n i q u e s are n o w b e i n g successfully a p p l i e d to r e l a t i v e l y small b r i d g e d electron-transfer m o d e l c o m p o u n d s (23-25) a n d i d e a l i z e d systems (26). O u r approach has r e l i e d o n o n e - e l e c t r o n and effective p o t e n t i a l methods because these m e t h o d s are adequate for addressing issues o f t u n n e l i n g energy d e p e n d e n c e a n d b r i d g e topology ef fects a n d because i t is possible to p e r f o r m these calculations i n v e r y w e a k l y c o u p l e d systems w i t h o u t serious c o n c e r n about basis set artifacts. Q u a l i t a t i v e issues r e l a t e d to t h r o u g h - b o n d a n d through-space c o u p l i n g are a d d r e s s e d c o n v e n i e n t l y w i t h carefully p a r a m e t e r i z e d exactly soluble square b a r r i e r models (27). T h e generic results o f the b r i d g e studies are s u m m a r i z e d i n F i g u r e 1. M o s t bridges can b e " r e d u c e d " to chains o f i n t e r a c t i n g pairs o f orbitals w i t h t w o characteristic interactions. T h e details of the r e d u c e d orbitals are d e t e r m i n e d b y the topology o f the c h a i n a n d energetics of the b o n d s i n t h e b r i d g e (28). T u n n e l i n g t h r o u g h a b r i d g e of s u c h r e p e a t i n g units w h e r e the m i x i n g i n t o the b r i d g e is weak a n d decay is r a p i d e n o u g h (decay p e r b r i d g e u n i t s q u a r e d is s m a l l c o m p a r e d to 1, not a v e r y stringent condition) allows Ηη, to b e w r i t t e n as i n e q 10. W r i t i n g the decay of Η p e r b o n d as e (12, 28-32) φ

N e g l e c t i n g backscattering b e t w e e n bonds,

e

. » Γ '

[(Ε

1

È* -

OaXE -

ct ) iR

(10b)

β, J 2

F o r |e| > 0.4, corrections for backscattering m u s t b e i n c o r p o r a t e d i n the calculation of e i t s e l f (12, 31). H e r e , Ε is the t u n n e l i n g energy, L a n d R refer to the left a n d r i g h t h y b r i d atomic orbitals i n the b o n d s , (N 4- 1) is the total n u m b e r o f b o n d s i n the b r i d g e , β is the i n t e r a c t i o n w i t h i n b o n d s , 7 is the i n t e r a c t i o n b e t w e e n b o n d s , α is the e n e r g y o f the (hybrid) orbitals i n the b o n d s , a n d β a n d β are the c o u p l i n g m a t r i x e l e m e n t s b e t w e e n the d o n o r a n d acceptor a n d the first a n d last b r i d g e units, r e s p e c t i v e l y . A s a n e x a m p l e , i n a l i n e a r e x t e n d e d h y d r o c a r b o n c h a i n 7/β — 0.25 a n d β — - 9 eV. E q u a t i o n s 10a a n d 10b are g e n e r a l i z e d i n the next section for the case i n w h i c h the b o n d types i n the b r i d g e may be c h e m i c a l l y different. β

Α

M o s t o f the electron-transfer m o d e l c o m p o u n d s a i m e d at t e s t i n g the distance d e p e n d e n c e of the transfer rate are o f the f o r m D B A , w h e r e η is variable. T h e p o t e n t i a l i n such l i n k e r s is, to a good a p p r o x i m a t i o n , p e r i o d i c (12, 28-30). T h e b o u n d a r y conditions o n the p e r i o d i c p o t e n t i a l c o n t a i n t h e details o f the d o n o r a n d acceptor s t r u c t u r e , b u t the p e r i o d i c n a t u r e o f the n


78


b r i d g e allows r e l a t i v e l y s i m p l e calculations to make p r e d i c t i o n s about t h e e n e r g y a n d s y m m e t r y d e p e n d e n c e of the c o u p l i n g w i t h i n b r o a d classes o f l i n k e r s . T h e s e p r e d i c t i o n s , w h i c h t y p i c a l l y i n c l u d e m o r e details t h a n w e r e u s e d to calculate eqs 10a a n d 10b, are r e l i a b l e as l o n g as t h e decay w i t h i n the b r i d g e is sufficiently r a p i d a n d the net m i x i n g onto the b r i d g e is weak. P r e d i c t i o n s for σ - b o n d - c o u p l e d e l e c t r o n transfer i n c l u d e d p o i n t i n g out t h e e n h a n c e d m e d i a t i o n properties of bridges w i t h convergent pathways of e q u a l l e n g t h , such as exist i n corner-fused rings (vs. edge-fused rings) a n d o t h e r effects (28-30). A l t h o u g h the theoretical calculations s e e m to be i n fair agree m e n t w i t h e x p e r i m e n t , there are several questions b e g g i n g to be a d d r e s s e d


synthetically. 1. F o r fixed reaction free energy, A G , b u t donor a n d acceptor energies v a r i e d i n an absolute sense, w i l l the decay l e n g t h of iirp change the p a r a m e t e r β / 2 i n is the d o n o r - a c c e p t o r

°c β χ ρ [ - Ι Ι β / 2 ] (where R

separation distance)? D o e s a hole or

electron-transfer m e c h a n i s m d o m i n a t e i n c h e m i c a l systems?

J

0.0

10.0

E n e r g y (eV) Figure la. e (decay per bond) vs. Ε plots are shown for a C-C chain with β = - 8 . 5 , α c = 0, and y = -2.2 eV. The infinite chain result (U-shaped curve) is shown (28, 29), as well as the hole- and the electron-mediation limits (eq 13). The approximate curves are adequate in energetic regimes expected for typical model compounds (|e| ~ 0.4-0.6). c


5.



79

2. F o r fixed d o n o r a n d acceptor b u t v a r i e d b r i d g e , w i l l the net c o u p l i n g show the p r e d i c t e d topological effects

(28-30)?

3. I n saturated systems c o u p l i n g i r - d o n o r s , does σ or IT s y m m e t r y c o u p l i n g into the b r i d g e dominate the net i n t e r a c t i o n , Η ? φ


4. H o w i m p o r t a n t are h y d r o g e n bonds for m e d i a t i n g e l e c t r o n transfer? S u r e l y there is a role for m o d e l b u i l d i n g h e r e . Is the p i c t u r e o f h y d r o g e n bonds as p r e f e r e n t i a l l y assisting h o l e m e diation (12) accurate? 5. H o w costly are the s y m m e t r y demands of σ/ττ interactions i n proteins? D o IT groups assist transfer or not? O u r c u r r e n t t h i n k i n g is that the π systems must be a l i g n e d i n special ways for significant enhancements. 6. T h e distance d e p e n d e n c e of A G a n d λ (reorganization energy) complicate the i n t e r p r e t a t i o n of b r i d g e a n d t u n n e l i n g e n e r g y d e p e n d e n c e studies because these parameters cannot be h e l d fixed w i t h transfer distance. C a n A G a n d λ studies b e p e r -

-10.0

o.o

-5.0

5.0

10.0

E n e r g y (eV) Figure lb. The energy dependence of e is shown for a C-C vs. C-N chain with β = -8.5, α Ν = - 3 . 3 , y = - 3 . 1 , and y = -2.2 eV. The C-N plot is centered at lower energy because of the greater electron affinity of nitrogen. The "U" shape of the curves is characteristic, where e shows distinct electronand hole-mediation regimes. N

c


80

E T IN INORGANIC, O R G A N I C , A N D BIOLOGICAL SYSTEMS

f o r m e d as a f u n c t i o n o f distance to u n a m b i g u o u s l y d e c o n v o l u t e the b r i d g e structure d e p e n d e n c e of the rates? A n s w e r s to these questions are w i t h i n synthetic a n d spectroscopic r e a c h , b u t o b t a i n i n g t h e m w i l l r e q u i r e a c o o r d i n a t e d effort. T h e m o d e l c o m p o u n d s a n d the t h e o r e t i c a l studies have taught us about the t y p i c a l l e n g t h scales for decay of t u n n e l i n g interactions i n saturated a n d unsaturated organic b r i d g e s . C o u n t i n g bonds along the shortest p a t h f r o m d o n o r to acceptor i n w e l l - c h a r a c t e r i z e d m o d e l c o m p o u n d s suggests that the decay o f Η is about a factor o f 0 . 4 - 0 . 6 p e r b o n d . Reference 3 s u m m a r i z e s e x p e r i m e n t a l l y m e a s u r e d values o f these parameters a n d t h e i r d e p e n d e n c e o n structural details o f the b r i d g e . W e have l e a r n e d from the t h e o r y that " d e c a y p e r b o n d " strategies w o r k r a t h e r w e l l for these t y p i c a l decays, w i t h some qualifications (12, 28-30). T h e propagation of the d o n o r a n d acceptor states can b e b u i l t u p b y s e q u e n t i a l l y i n t r o d u c i n g single bonds (or groups o f bonds i n strongly d e l o c a l i z e d systems) to the c h a i n o f orbitals. F o l l o w i n g the a d d i t i o n of each b o n d , the a m p l i t u d e l e a k i n g onto it is c a l c u l a t e d as a 2 x 2 p r o b l e m . Interference effects can b e treated w i t h i n this strategy (12, 28-32) i f i n t e r s e c t i n g pathways b e a r i n g s i m i l a r a m p l i t u d e s are h a n d l e d c a r e fully.


φ

Protein-Mediated Electron Transfer Interpreted with Decay-per-Bond Methods A l t h o u g h i n t r i g u i n g questions r e m a i n i n the m o d e l c o m p o u n d area, o u r a i m i n p u r s u i n g that w o r k was to l e a r n h o w to piece together a n d p a r a m e t e r i z e a m o d e l for p r o t e i n - m e d i a t e d e l e c t r o n transfer i n p h o t o s y n t h e t i c a n d r e s p i r a t o r y reactions. A physical tunneling pathway is d e f i n e d as a c o l l e c t i o n o f i n t e r a c t i n g bonds i n a p r o t e i n a r o u n d a n d b e t w e e n the d o n o r a n d acceptor that m a k e some c o n t r i b u t i o n to the d o n o r - a c c e p t o r i n t e r a c t i o n . A few spe cific p h y s i c a l pathways m a y or m a y not d o m i n a t e the electronic c o u p l i n g b e t w e e n d o n o r a n d acceptor. W h e t h e r a r e l a t i v e l y small n u m b e r o f pathways is adequate to d e s c r i b e the c o u p l i n g i n proteins is actually a d e e p t h e o r e t i c a l issue. W e argue (on the basis o f r a p i d decay o f through-space interactions for t y p i c a l t u n n e l i n g energies, the r e l a t i v e l y l o w d e n s i t y o f residues, a n d the anisotropic p a c k i n g o f bonds) that a r e l a t i v e l y s m a l l n u m b e r of pathways is l i k e l y to b e i m p o r t a n t . T h e d e c a y - p e r - b o n d approach leads to e q 11 for the c o n t r i b u t i o n to the t u n n e l i n g m a t r i x e l e m e n t arising f r o m a single p h y s i c a l p a t h w a y w i t h N covalent c o u p l i n g s b e t w e e n b r i d g e bonds, N through-space contacts, a n d N h y d r o g e n bonds (12, 31, 32) B

s

B

β βοβ, Α

N

»

N s

N e


5.


From Model Compounds

to Proteins

81

βϋ β λ couples the d o n o r or acceptor i n t o the first o r last b o n d of the b r i d g e , respectively; f is the c o n t r i b u t i o n to a r i s i n g from a single pathway; a n d β is the c o u p l i n g b e t w e e n orbitals i n the first b o n d . Values for e can often be a p p r o x i m a t e d b y u s i n g p e r t u r b a t i o n theory. A s an e x a m p l e , the lowest o r d e r c o n t r i b u t i o n to e is g i v e n b y e q 10b. T h i s l i m i t totally neglects backscattering b e t w e e n bonds, a n d corrections to it n e e d to b e i n c l u d e d for large €. F o r a p a r t i c u l a r i n t e r a c t i o n , e can b e dissected i n t o c o n t r i b u t i o n s from e l e c t r o n a n d h o l e m e d i a t i o n across a b o n d (12) as o

r

da

χ

e = e

+


e

(12)

e

h

I n the l i m i t w h e r e hole m e d i a t i o n t h r o u g h the b o n d d o m i n a t e s , for e x a m p l e , and the two c o u p l e d covalent b o n d s are the same

€ =

= —

(13)

J

&

Electron Transfer in Inorganic, Organic, and Biological Systems

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