Polymers in Electronics - American Chemical Society

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34 Redox Properties of Conjugated Polymers A Successful Correlation of Theory and Experiment R. R. CHANCE and D. S. BOUDREAUX Allied Corporation, Morristown, NJ 07960 J. L.BREDAS1and R. SILBEY Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139 The electrochemical properties of conductive polymer systems are important with regard to understanding the electrochemical doping process and in applications of conductive polymers as battery electrodes. We have developed a computa tional method, based on the Valence Effective Hamiltonian technique, which is remarkably effective in the computation of oxidation and reduction potentials of a variety of conjugated polymers (polyacetylene, polyphenylene, polythiophene, polypyrrole) and their oligomers. For example, the VEH results for polyacetylene yield an oxidation potential of 0.4 volts versus SCE and a reduction potential of -1.1 volts versus SCE, both of which are in good agreement with experiment. Similar agreement is found with the other systems. Of special importance is the successful prediction of the surprisingly low oxidation potential of polypyrrole (-0.4 volts versus SCE). A number of organic polymers become electrically conducting on addition of electron donors or acceptors.(1-5) Despite the enor mous interest in these conducting polymer systems, many theore tical aspects of the problem remain poorly understood, especially with regard to the electronic properties of the "doped" (partially ionized) polymers. Progress is being made, however, in understanding the undoped polymer precursors. In a series of recent papers, we have demonstrated the utility of the 1Current address: Lab. Chemie Theorique Appliquee Fac. Univ. NOTRE-DAME de la Paix, Namur, Belgium. 0097-6156/84/0242-0433$06.00/0 © 1984 American Chemical Society In Polymers in Electronics; Davidson, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

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V a l e n c e E f f e c t i v e H a m i l t o n i a n (VEH) method i n d e s c r i b i n g t h e ground s t a t e p r o p e r t i e s o f conjugated p o l y m e r s — i n p a r t i c u l a r t h o s e w h i c h become h i g h l y c o n d u c t i n g upon doping»(6-9) The VEH method e m p l o y s a t o m i c p o t e n t i a l s d e r i v e d f r o m d o u b l e z e t a q u a l i t y ab i n i t i o c o m p u t a t i o n s on s m a l l m o l e c u l e s i n c a l c u l a t i o n s on l a r g e m o l e c u l e s . W i t h t h i s m e t h o d , x - r a y p h o t o e l e c t r o n s p e c t r a , i o n i z a t i o n p o t e n t i a l s , a n d o p t i c a l band gaps f o r a number o f p o l y m e r s have been computed i n good agreement with experiment.(6-9) The s u c c e s s f u l c a l c u l a t i o n o f p o l y m e r i o n i z a t i o n p o t e n t i a l s i s e s p e c i a l l y important, since the i o n i z a t i o n p o t e n t i a l determines whether o r not a p a r t i c u l a r a c c e p t o r dopant w i l l be c a p a b l e o f i o n i z i n g t h e p o l y m e r . I n t h i s p a p e r we p r e s e n t VEH c a l c u l a t i o n s on o l i g o m e r s o f polyacetylene (PA), poly(p-phenylene) (PPP), polythiophene ( P T P ) , a n d p o l y p y r r o l e (PPY) a n d compare t h e r e s u l t s t o VEH polymer c a l c u l a t i o n s . F o r PTP a n d PPY, c o n s i d e r a b l e u n c e r t a i n t y e x i s t s f o r t h e i r g e o m e t r y - p a r t i c u l a r l y i n the carbonc a r b o n bonds between monomer u n i t s . F o r t h i s r e a s o n we have e m p l o y e d a s e m i e m p i r i c a l quantum m e c h a n i c a l t e c h n i q u e d e s i g n e d f o r geometry p r e d i c t i o n , MNDO ( M o d i f i e d N e g l e c t o f D i f f e r e n t i a l O v e r l a p ) 1 0 . The MNDO p r e d i c t e d geometry i s u s e d a s i n p u t t o the VEH p r o g r a m . Our c a l c u l a t i o n s c o n c e n t r a t e on o p t i c a l "band g a p s " a n d i o n i z a t i o n p o t e n t i a l s a n d use o p t i c a l a n d e l e c t r o c h e m i c a l d a t a f o r c o m p a r i s o n . We e m p h a s i z e t h e s u c c e s s f u l p r e d i c t i o n o f e l e c t r o c h e m i c a l o x i d a t i o n and r e d u c t i o n p o t e n t i a l s , which i s e s p e c i a l l y important c o n s i d e r i n g recent a c t i v i t y i n the a p p l i c a t i o n o f conjugated polymers i n r e c h a r g e a b l e b a t t e r i e s H » 12. T h e o r e t i c a l and Computational

Technique

I n t h i s s e c t i o n we d i s c u s s b r i e f l y t h e VEH t e c h n i q u e a n d o u r a p p r o a c h t o t h e p r o b l e m i n c l u d i n g t h e u s e o f MNDO a s a geometry i n p u t d e v i c e . The m e t h o d o l o g y f o r o b t a i n i n g m o l e c u l a r o n e - e l e c t r o n H a m i l t o n ! a n s f r o m f i r s t p r i n c i p l e s h a s been w o r k e d o u t by N i c o l a s a n d D u r a n d . ( 1 3 ) The e f f e c t i v e Fock o p e r a t o r o f t h e m o l e c u l e i s assumed t o be t h e sum o f t h e k i n e t i c energy and the v a r i o u s atomic p o t e n t i a l s i n the molecule: F eff

= - A 2

+ J A

VA

where V^ i s t h e e f f e c t i v e p o t e n t i a l o f atom A. F o r computational ease, simple n o n l o c a l atomic p o t e n t i a l s chosen o f the form o f Gaussian p r o j e c t o r s :

(1)

are

In Polymers in Electronics; Davidson, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

34.

Redox Properties of Conjugated

C H A N C E ET AL.

V

A -

I I

I

l m

ij

C

1

J

f

t

o

| yk

yk

iim

Polymers

J

435

(2)

jto

where the summations over I and m define the angular dependence of ν^· The numerical c o e f f i c i e n t s C i j ^ are independent of m i n the case of s p h e r i c a l symmetry, which we u s u a l l y c o n s i d e r . The functions are normalized Gaussians:


XiAm -

N

±

βχp[-αΐΓ ]Υ^(0,φ)

(3)

2

% i s the n o r m a l i z a t i o n f a c t o r and denotes the usual s p h e r i c a l harmonics. Note that only Is and 2p Gaussian C a r t e s i a n functions are used. The parameterizations of the l i n e a r c o e f f i c i e n t s , C., and the n o n l i n e a r exponents, a, f i r s t require valence SCF c a l c u l a t i o n s on model molecules by a t h e o r e t i c a l pseudopotential method(14) with an ST0-3G minimal basis set and a double zeta basis s e t . The model molecules chosen to parameterize carbon and hydrogen atomic p o t e n t i a l s were ethane, transbutadiene, and a c e t y l e n e ( 6 ) ; f o r s u l f u r and carbon l i n k e d to s u l f u r , dimethyl s u l f i d e and thiophene(jB); for n i t r o g e n and carbon l i n k e d to n i t r o g e n , dimethylamine and p y r r o l e . ( 1 5 ) For each molecule, the Fock operator i s constructed a s : F

- l

εν

I

Φν

Φν

I

4

υ where the summation i s over a l l occupied s t a t e s ; the valence o r b i t a l e φ are taken from the minimal basis set c a l c u l a t i o n and the corresponding monoelectronic energies ε from the double zeta c a l c u l a t i o n . The choice of t h i s t h e o r e t i c a l Fock operator leads to valence e f f e c t i v e Hamiltonians p r o v i d i n g double zeta accuracy for monoelectronic energies when solved with a minimal s e t . The parameterization of the atomic p o t e n t i a l s i s then determined by minimizing the q u a n t i t y υ

υ

I (F-F | F - F Wecule Β (5) Β where the summation runs over the model molecules used for a given set of atomic p o t e n t i a l s . ( F - F f f | F - F f f ) denotes the s c a l a r product of F - F f f with i t s e l f i n the subspace of the occupied valence o r b i t a l s . ( 1 3 ) On the model molecules, standard d e v i a t i o n s between the ε energies produced by using the valence e f f e c t i v e Hamiltonians and double zeta energies are of the order of 0.015 a . u . , and i n no case l a r g e r than 0.2 a . u . ( 6 , 8 , 1 5 ) . The p o s i t i o n of the highest occupied o r b i t a l e f f

e f f

e

e

e

υ



436


i s e s p e c i a l l y well-reproduced - a r e s u l t which lends confidence i n o b t a i n i n g good i o n i z a t i o n p o t e n t i a l e s t i m a t e s . No i n f o r m a t i o n p e r t a i n i n g t o t h e e x c i t e d s t a t e s i s i n c l u d e d i n t h e a t o m i c p o t e n t i a l s . As a r e s u l t , no s p e c i a l a t t e n t i o n s h o u l d i n p r i n c i p l e be g i v e n t o t h e u n o c c u p i e d l e v e l s . However, f o r t h e p l a n a r s y s t e m s c o n s i d e r e d p r e v i o u s l y , s u r p r i s i n g l y good agreement between e x p e r i m e n t and t h e o r y has been o b t a i n e d f o r the lowest o p t i c a l energy t r a n s i t i o n s . ( 7 ) The e x t e n s i o n of t h e VEH method t o p o l y m e r c a l c u l a t i o n s i s straightforward(6,16)· The e f f e c t i v e o p e r a t o r t a k e s t h e f o r m : Fef f » - Τ + ï I V g A

(6)

A

t h e summations o v e r g and A r u n n i n g , r e s p e c t i v e l y , o v e r t h e p o l y m e r u n i t c e l l s and t h e atoms p r e s e n t i n one c e l l . The band s t r u c t u r e E ( k ) o f t h e p o l y m e r , where k i s a p o i n t i n t h e f i r s t B r i l l o u i n zone o f t h e p o l y m e r , i s o b t a i n e d f r o m e i g e n v a l u e s o f t h e s e t of s e c u l a r e q u a t i o n s : F(k) Ç(k)

= j5(k)

C(k)

E(k)

(7)

j F ( k ) and j>(k) a r e t h e F o c k and o v e r l a p m a t r i c e s between B l o c h f u n c t i o n s and £ ( k ) c o l l e c t s t h e c o e f f i c i e n t s o f t h e l i n e a r com b i n a t i o n s of B l o c h f u n c t i o n s t h a t p r o v i d e the c r y s t a l l i n e o r b i tale. The m a i n a d v a n t a g e s o f t h e VEH t e c h n i q u e a r e t h a t i t i s c o m p l e t e l y t h e o r e t i c a l and g i v e s ab i n i t i o d o u b l e z e t a q u a l i t y r e s u l t s w i t h n e g l i g i b l e computer t i m e , s i n c e o n l y o n e - e l e c t r o n i n t e g r a l s need t o be e v a l u a t e d and SCF i t e r a t i v e c y c l e s a r e completely avoided. I t must be p o i n t e d o u t t h a t t h e VEH a t o m i c p o t e n t i a l s have n o t been p a r a m e t e r i z e d f o r g e o m e t r y o p t i m i z a t i o n p u r p o s e s and s h o u l d be u s e d w i t h g e o m e t r i c p a r a m e t e r s c l o s e t o e q u i l i b r i u m . F o r s y s t e m s whose g e o m e t r i e s a r e e x p e r i m e n t a l l y unknown (as i s t h e c a s e o f t h e m a j o r i t y o f t h e l a r g e o l i g o m e r s and p o l y m e r s s t u d i e d i n t h i s p a p e r ) , we must make u s e o f o t h e r t e c h n i q u e s i n order to obtain reasonable input geometries. Ab i n i t i o t e c h n i q u e s , e v e n w i t h s m a l l b a s i s s e t s , r a p i d l y become t o o e x p e n s i v e when l a r g e compounds a r e c o n s i d e r e d . As a r e s u l t , we have c h o s e n t o o p t i m i z e t h e g e o m e t r i e s o f a l l o l i g o m e r s and p o l y m e r s s t u d i e d h e r e w i t h t h e MNDO ( M o d i f i e d N e g l e c t o f D i f f e r e n t i a l Overlap) semiempirical procedure.(10) T h i s method has been t h o r o u g h l y t e s t e d on o r g a n i c compounds c o n t a i n i n g c a r b o n , h y d r o g e n , n i t r o g e n , o x y g e n , and s u l f u r and r e p r o d u c e s f a i r l y w e l l the experimental geometries.(17) We f i n d t h a t t h e VEH r e s u l t s a r e q u a l i t a t i v e l y u n a f f e c t e d by s m a l l d i f f e r e n c e s i n i n p u t g e o m e t r i e s . This is i l l u s t r a t e d


34.

C H A N C E ET AL.


Polymers

437


f o r p o l y p y r r o l e i n T a b l e I where we p r e s e n t t h e VEH i o n i z a t i o n p o t e n t i a l s a n d band gaps f o r f o u r d i f f e r e n t g e o m e t r i e s : t h e MNDO o p t i m i z e d g e o m e t r y , t h e ST0-3G o p t i m i z e d g e o m e t r y ( 1 8 ) , and t h e e x p e r i m e n t a l g e o m e t r y o f p y r r o l e ( 1 9 ) w i t h i n t e r - r i n g bond l e n g t h s o f 1.45A and 1.49A. The i o n i z a t i o n p o t e n t i a l v a l u e s f a l l between 5.68 eV a n d 5.98 eV a n d band gaps between 3.0 eV and 4.0 eV.

TABLE I :

E v a l u a t i o n o f t h e VEH i o n i z a t i o n p o t e n t i a l ( I P ) and band gap ( E ) v a l u e s f o r p o l y p y r r o l e a s a f u n c t i o n o f t h e g e o m e t r y : MNDO o p t i m i z e d g e o m e t r y ( t h i s w o r k ) , ST0-3G o p t i m i z e d g e o m e t r y ( 1 8 ) , a n d e x p e r i m e n t a l r i n g geometry ( f o r p y r r o l e ) ( 1 9 ) w i t h i n t e r - r i n g bond l e n g t h s o f 1.45A and 1.49A. g

MONOMER EXP.

MNDO

ST0-3G

RC-N(A)

1.399

1.385

1.38

1.38

RcrcU)

1.410

1.363

1.37

1.37

Rc-c(A)

1.426

1.420

1.43

1.43

C-N-C(°)

110.8

109.2

109-0

109.0

C-C-N(°)

106.5

107.4

108.0

108.0

i n t e r - r i n g ( A ) 1.453

1.474

1.45

1.49

IP ( e V )

5.68

5.96

5.82

5.98

Eg ( e V )

3.0

3.9

3.6

4.0

geometry:

R

R e s u l t s and D i s c u s s i o n Our r e s u l t s a r e s u m m a r i z e d i n T a b l e I I . I n c l u d e d i n t h e t a b l e a r e VEH computed i o n i z a t i o n p o t e n t i a l s ( I P ) a n d band gaps ( E g ) , a n d e x p e r i m e n t a l band gaps a n d gas phase i o n i z a t i o n p o t e n t i a l s when a v a i l a b l e . We have shown p r e v i o u s l y t h a t t h e I P v a l u e s f r o m VEH t h e o r y f o r t h e p o l y m e r s , a f t e r s u b t r a c t i n g ~1.9 eV f o r a s o l i d s t a t e p o l a r i z a t i o n c o r r e c t i o n , a r e i n good agreement w i t h s o l i d - s t a t e I P v a l u e s . From T a b l e I I t h e I P


438

p o l y m e r s in e l e c t r o n i c s


values for monomers and dlmers are also seen to be In reasonable agreement with gas phase values. The agreement of VEH theory and experiment for optical band gaps i s also quite good* (For convienence, we use the term "bandgap" even when referring to the optical absorption peak of the oligomers.) As we have noted previously, no attempt has been made to design VEH theory to treat excited states. Thus the relatively good agreement between theory and experiment evident i n Table II and i n previous work(7) i s unexpected. TABLE II; Gas Phase Ionization Potentials (IP) and Band Gaps (Eg). A l l energies are given i n eV.

Chain Length

System

Polyacetylene (PA)

1 2 3 00

Poly(p-phenylene) (PPP)

1 2 3 00

Polythiophene (PTP)

1 2 3 00

Polypyrrole (PPY)

1 2 3 00

Experiment IP

VEH IP

H

Ref.

5.7 4.6 1.8

20 20 ,21 21 22

9.2 8.3 8.2

5.9 4.9 4.4 3.6

23 ,24 23 ,24 23 ,24 2

5.51 3.65 2.93 1.71

9.0

5.4 4.1 3.5

20 ,24 24 24

7.01 5.24 4.35 2.99

8.2

6.0 4.4 3.6 3.0

25 ,24 25 ,24 25 ,24 1

10.16 8.78 8.12 6.67

7.85 5.29 4.09 1.45

10.5 9.1

9.30 8.34 7.97 7.45

6.69 4.88 4.18 3.23

9.29 8.14 7.71 7.01

8.13 6.89 6.43 5.68

In Figure 1, we have plotted the experimental and theoretical data for the band gap of the polyenes versus l/n. A linear relationship i s expected for large n. Also included i n Figure 1 are data for diphenylpolyenes (DPP), C5H5-(HC=CH) -C6H5, which are more stable than polyenes, H-(HC=CH) -H; consequently, there X

n


34.


C H A N C E ET AL.

439

Polymers

CONJUGATION LENGTH (n)


oo

10

5

ι

ι

4

3

2

1

11-

0.0

ι

ι

I

ι

ι

ι

ι

0.5

1.0

1/n

Figure

1.

Band gap f o r

p o l y e n e s , ( 2 1 ) H(HC=CH) H and n

diphenylpolyenes (DPP),(27,28) C6H -(HC=CH) -C6H5, plotted against reciprocal conjugation length. The h i g h e s t band gap DPP r e s u l t i s f o r s t i l b e n e ( x = l ) ; t h e l o w e s t DPP r e s u l t i s f o r x=7. For DPP, η i s the e f f e c t i v e c o n j u g a t i o n l e n g t h computed as x + 2 . 7 . A l l e x p e r i m e n t a l d a t a r e f e r t o a b s o r p t i o n peaks i n nonpolar organic s o l v e n t s . 5

X


440


i s more e x t e n s i v e l i t e r a t u r e ( 2 6 - 2 8 ) on DPP m o l e c u l e s . When c o n s i d e r i n g DPP as a model f o r t h e p o l y a c e t y l e n e s y s t e m , t h e e f f e c t o f t h e p h e n y l end g r o u p s on t h e e l e c t r o n i c p r o p e r t i e s must be c o n s i d e r e d . A c c o r d i n g l y , we d e f i n e a n e f f e c t i v e c o n j u g a t i o n l e n g t h , neff=x+A, so t h a t A d e s c r i b e s t h e e x t e n s i o n o f t h e c o n j u g a t i o n l e n g t h by t h e p h e n y l end g r o u p s beyond t h e x u n i t p o l y e n e s e q u e n c e . The v a l u e o f A i s e s t a b l i s h e d by a d j u s t i n g i t u n t i l t h e e x p e r i m e n t a l band gap o f DPP w i t h χ d o u b l e bonds i s e q u a l t o t h a t o f a p o l y e n e w i t h x+A d o u b l e bonds.29 The r e s u l t s shown i n F i g u r e 1 u s e A 2 . 7 and p r o v i d e an e x c e l l e n t c o r r e s p o n d e n c e between t h e p o l y e n e and d i p h e n y l p o lyene data. We have u s e d t h i s a p p r o a c h p r e v i o u s l y t o show t h a t the three o p t i c a l t r a n s i t i o n energies observed f o r r a d i c a l a n i o n s i n DPP e x t r a p o l a t e t o y i e l d a good d e s c r i p t i o n o f t h e n e a r - i n f r a r e d ("mldgap") a b s o r p t i o n f o u n d w i t h d o n o r - d o p i n g o f polyacetylene.(29) VEH c a l c u l a t i o n s have b e e n c a r r i e d o u t f o r two DPP m o l e c u l e s ( x = l and x = 2 ) . As c a n be c o n c l u d e d f r o m F i g u r e 1, t h e VEH computed band gaps a r e i n r e a s o n a b l e agreement w i t h t h e p o l y e n e d a t a . I P r e s u l t s a r e s u m m a r i z e d i n F i g u r e 2. F i r s t n o t e t h a t t h e t h e o r e t i c a l I P v a l u e s f o r t h e two DPP m o l e c u l e s a r e i n n e a r p e r f e c t agreement w i t h t h o s e d e r i v e d f r o m VEH p o l y e n e v a l u e s w i t h u s e o f t h e same A 2 . 7 c o r r e c t i o n f a c t o r n o t e d a b o v e . I t i s a l s o c l e a r from F i g u r e 2 t h a t t h e o r e t i c a l IP values f o r t h e DPP s y s t e m , i n c l u d i n g t h o s e i n f e r r e d f r o m t h e p o l y e n e r e s u l t s , a r e i n e x c e l l e n t agreement w i t h t h e e x p e r i m e n t a l I P d a t a o f Hudson e t a l . ( 3 0 ) The agreement i s as good as t h a t o b t a i n e d w i t h t h e CND0/S2 s e m i e m p i r i c a l p r o c e d u r e , 3 1 b u t w i t h o u t any a d j u s t m e n t o f e n e r g y s c a l e s a n d , more i m p o r t a n t l y , w i t h o u t any input of experimental i n f o r m a t i o n (except, of course, the g e o m e t r y ) . I n f a c t , i n c l u d i n g o u r MNDO g e o m e t r y d e t e r m i n a t i o n , t h e e x c e l l e n t agreement w i t h e x p e r i m e n t d i s p l a y e d i n F i g u r e s 1 and 2 and T a b l e I I i s o b t a i n e d w i t h i n p u t o f o n l y t h e atom connectivity. A p r i n c i p l e r e a s o n f o r c o n s i d e r i n g t h e DPP m o l e c u l e s i n s u c h d e t a i l i s t h a t a f a i r l y c o m p l e t e s e t o f o x i d a t i o n ( 3 2 ) and r e d u c t i o n p o t e n t i a l s ( 2 6 ) i s a v a i l a b l e . We a r e i n t e r e s t e d i n t h e e x t e n t t o w h i c h s u c h d a t a c a n be p r e d i c t e d t h e o r e t i c a l l y and i n a p p l y i n g t h e o r y and oligomer e x t r a p o l a t i o n s t o understanding t h e e l e c t r o c h e m i c a l p r o p e r t i e s of polyacetylene. The l a t t e r i s e s p e c i a l l y important because of t h e l a r g e i n t e r e s t i n b a t t e r y a p p l i c a t i o n s of p o l y a c e t y l e n e and o t h e r conjugated polymers.(12) I n F i g u r e 3, we have p l o t t e d t h e t h e o r e t i c a l v a l u e s o f t h e i o n i z a t i o n p o t e n t i a l s f o r t h e p o l y e n e s a n d DPP v e r s u s l / n ; on t h e same p l o t a r e e x p e r i m e n t a l o x i d a t i o n p o t e n t i a l s ( E ) f o r DPP. An a p p r o x i m a t e l y l i n e a r c o r r e l a t i o n between I P and E i s e x p e c t e d . ( 3 3 ) F u r t h e r m o r e , we have p l o t t e d t h e t h e o r e t i c a l


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