Bond Length Alternation and Forbidden Energy Gap in Conjugated

t h e o r e m (2^) ... chains with partly filled energy bands (metallic spectrum) are ... cell, E t , as function of the asymmetric distortion, A , of...
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9 Bond Length Alternation and Forbidden Energy Gap in Conjugated Periodic Polymers M. KERTESZ

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Central Research Institute for Chemistry, Hungarian Academy of Sciences, 1525 Budapest, Pf. 17, Hungary J. K O L L E R and A. A Z M A N B . Kidric Chemical Research Institute, 61001 Ljubljana, P.p. 380, Yugoslavia

The N u c l e a r D i s t o r t i o n P r o b l e m I n r e c e n t y e a r s study o f q u a s i o n e - d i m e n s i o n a l (ID) systems has become an i n c r e a s i n g l y i m p o r t a n t f i e l d o f s o l i d s t a t e p h y s i c s and c h e m i s t r y (1_). One b a s i c i d e a h e r e i s the P e i e r I s t h e o r e m (2^) s t a t i n g t h a t e q u i d i s t a n t n u c l e a r c o n f i g u r a t i o n s o f s t r i c t l y ID c h a i n s w i t h p a r t l y f i l l e d e n e r g y bands ( m e t a l l i c s p e c t r u m ) are u n s t a b l e towards n u c l e a r d i s t o r t i o n s l e a d i n g to f o r b i d d e n energy gaps i n t h e o n e - p a r t i c l e s p e c t r u m . There i s another i n s t a b i l i t y o f a q u i t e d i f f e r e n t k i n d (_3) p r e s e n t i n t h e H a r t r e e - F o c k - t y p e w a v e f u n c t i o n o f m e t a l l i c (and sometimes also other) systems: i n t r o d u c t i o n o f c e r t a i n e x t r a v a r i a t i o n a l freedom ( u s u a l l y f o r m a l l y i n c r e a s i n g the l a t t i c e p e r i o d ) may l e a d t o o c c u r r e n c e o f symmetry b r e a k i n g (SB) s o l u t i o n s s u c h as charge-density-waves (CDW) and s p i n - d e n s i t y - w a v e s (SDW). P e r f o r m i n g H a r t r e e - F o c k (HF) t y p e e l e c t r o n i c band s t r u c t u r e c a l c u l a t i o n s on a number o f ID p o l y m e r s ( 4 , _5, 6) i t has b e e n o b s e r v e d t h a t o c c u r r e n c e o f CDW s o l u t i o n s p o i n t s towards e n e r g e t i c a l advantageousness o f n u c l e a r distortions. Q u i t e r e c e n t l y C a l a i s , P u l a y (_7) and o t h e r s have drawn a t t e n t i o n to the s t u d y o f t h i s c o u p l i n g m e c h a n i s m . We examine the r e l a t i o n o f t h e s e b r o k e n symmetry HF w a v e f u n c t i o n s and the b e h a v i o r o f t h e p o t e n t i a l e n e r g y c u r v e o f t h e n u c l e i a r o u n d the e q u i d i s t a n t c o n f i g u r a t i o n . 1

The d e g r e e o f bond l e n g t h a l t e r n a t i o n i s c l o s e l y r e l a t e d to t h e m a g n i t u d e o f the f o r b i d d e n e n e r g y g a p , E , w h i c h i n t u r n a f f e Q t s e l e c t r i c a l , o p t i c a l and o t h e r p h y s i c a l p r o p e r t i e s o f t h e system. W i t h i n t h e o n e - e l e c t r o n p i c t u r e ( H u c k e l - m o d e l ) one c a n show, t h a t E = 2 | $ " $ | where 6 ^ i the resonance^ i n t e g r a l ! c o r r e s p o n d i n g t o t h e snoTfc and l o n g § o n d s i n an a l t e r n a t i n g p o l y e n e , r e s p e c t i v e l y . The o r i g i n o f t h e E i n p u r e p o l y a c e t y l e n e , (CH) , has been d i s c u s s e d r e p e a t e d l y ( 8 ) ^ a n d we p r e s e n t some t h e o r e t i c a l e v i d e n c e a t H a r t r e e - F o c k l e v e l , t h a t nonzero e x p e r i m e n t a l E i s c o m p a t i b l e w i t h the a l t e r n a t i n g m o d e l . A r e l a t e d p r o b l e m i s f%und a t t h e p o l y d i a c e t y l e n e s ( P D A ' s ) where t h e two d i f f e r e n t l y b o n d - a l t e r n a t e d structures have different absorption. a

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0097-6156/81/0162-0105$05.00/0 © 1981 American Chemical Society In Photon, Electron, and Ion Probes of Polymer Structure and Properties; Dwight, David W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

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PHOTON, ELECTRON, AND ION PROBES

F o r c e s A c t i n g On The N u c l e i The r e s u l t s o f a s t u d y o f t h e H u c k e l H a m i l t o n i a n e n a b l e d t h e f o l l o w i n g c l a s s i f i c a t i o n o f t h e shape o f t h e t o t a l e n e r g y p e r u n i t c e l l , E , as f u n c t i o n o f the a s y m m e t r i c d i s t o r t i o n , A , o f the chain: (i) E ^ l l l l (Peierls case); ( i i ) E ^ -x'A + y^A ( h e t e r o a t o m i c c h a i n ) , ( i i i ) E ^± x A + y^ A (bond-alternating case). The corresponding Huckel H a m i l t o n i a n s may be i l l u s t r a t e d a s : (i) B^j^B; ( i i ) ^ B ^ i B . ; and (iii) j^G'ij*. Case ( i ) i s l a b i l e , c a s e ( i i i ) p r e d i c t s n e c e s s a r i l y bond l e n g t h a l t e r n a t i o n , w h i l e i n c a s e ( i i ) t h e s t a b i l i t y o f the e q u i d i s t a n t c o n f i g u r a t i o n ( A = 0 ) depends on t h e s i g n o f y ' x . W i t h o u t d i s c u s s i n g more s o p h i s t i c a t e d H a m i l t o n i a n s , e x p l i c i t l y i n c l u d i n g Coulomb i n t e r a c t i o n l e t us d i s c u s s t h e e f f e c t o f c h a r g e d e n s i t y waves (CDW) h e u r i s t i c a l l y . Suppose t h a t t h e r e i s a CDW-type e l e c t r o n i c d i s t r i b u t i o n a d v a n t e g o u s f o r some o r o t h e r r e a s o n i n the c h a i n and i t s e f f e c t i s f e l t by the n u c l e i t h r o u g h Coulomb-forces. I n o r d e r to t a k e i n t o a c c o u n t t h i s e f f e c t w i t h i n t h e H u c k e l m o d e l we have t o change t h e a s a n d / o r 3 s . Traditiona l CDW c o r r e s p o n d s t o a l t e r n a t i n g + and - c h a r g e s on n e i g h b o r i n g atoms and c o n s e q u e n t l y n e i g h b o r i n g a v a l u e s change a l s o i n an alternating pattern. The a l t e r n a t i o n o f t h e r e s o n a n c e i n t e g r a l s , 3, c o r r e s p o n d to an o f f - d i a g o n a l CDW where c h a r g e s a l t e r n a t e i n t h e bond r e g i o n ( o f f - d i a g o n a l CDW o r b o n d - o r d e r a l t e r n a t i o n w a v e , BOAW). A s a c o n s e q u e n c e w i t h i n the H u c k e l m o d e l , i n c a s e o f BOAW, bond l e n g t h d i s t o r t i o n i s p r e d i c t e d w h i l e i n c a s e o f d i a g o n a l CDW t h i s depends on t h e r e l a t i o n o f x and y . F o r a q u a l i t a t i v e d i s c u s s i o n the H e l l m a n n - F e y n m a n t h e o r e m may be u s e d . T h e n , t h e f o r c e s a c t i n g on t h e n u c l e i a r e d e t e r m i n e d s o l e l y by the c l a s s i c a l e l e c t r o s t a t i c f o r c e s due t o the c h a r g e d i s t r i b u t i o n o f t h e e l e c t r o n s and n u c l e i . I n o r d e r t o see i f t h e x component ( a l o n g the c h a i n ) o f the f o r c e a c t i n g on n u c l e u s i , F^ i s z e r o o r n o t , i t i s s u f f i c i e n t t o c h e c k w h e t h e r the c h a r g e d i s t r i b u t i o n i s i n v a r i a n t u n d e r t h e x -> - x t r a n s f o r m a t i o n o r n o t . S i n c e i n e q u i d i s t a n t l i n e a r a r r a n g e m e n t the f i e l d o f the o t h e r n u c l e i i s i n v a r i a n t , i t i s n e c e s s a r y to c o n s i d e r t h e e l e c t r o n s only. From t h i s consideration solely follows that nonzero H e l l m a n n - F e y n m a n f o r c e s o c c u r i n t h e BOAW c a s e b u t n o t i n t h e d i a g o n a l CDW c a s e . This c o n c l u s i o n i s a l s o v a l i d for z i g z a g c h a i n s i f t h e n u c l e i a r e i n p l a n e s o f m i r r o r symmetry o r t h o g o n a l t o the x - a x i s , l i k e i n e q u i d i s t a n t p o l y a c e t y l e n e , (CH) and p o l y m e t h y n e i m i n e , (HCN) . From the t e c h n i c a l p o i n t o f v i e w i t i s i m p o r t a n t to n o t e , t h a t symmetry v i o l a t i n g i n c o r r e c t i n t e g r a l a p p r o x i m a t i o n s u s e d i n the l i t e r a t u r e (9) l e a d i n s e v e r a l c a s e s i n c o r r e c t l y t o BOAW a n d , c o n s e q u e n t l y , t o wrong p o t e n t i a l e n e r g y curve around A = 0. t

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Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 4, 2016 | http://pubs.acs.org Publication Date: August 13, 1981 | doi: 10.1021/bk-1981-0162.ch009

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In Photon, Electron, and Ion Probes of Polymer Structure and Properties; Dwight, David W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

9.

KERTESZ ET AL.

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Actual

Calculations

107

Conjugated Periodic Polymers ( H ) , (C> x

x >

( C H ) ^ and ( H C N ) ^ :

The m a i n r e s u l t s o f t h e c a l c u l a t i o n s a t ab i n i t i o H a r t r e e F o c k l e v e l may be s u m m a r i z e d , t h a t t h e e q u i d i s t a n t hydrogen, c a r b o n and (CH) c h a i n s h a v e , besides the m e t a l l i c s o l u t i o n a n o t h e r one w i t h ^ o w e r t o t a l e n e r g y p o s s e s s i n g a CDW. The l a t t e r i s o f the BOAW t y p e i n a l l t h r e e c a s e s , a n d c o n s e q u e n t l y t h e c o r r e s p o n d i n g t o t a l e n e r g y c u r v e s t a r t s l i n e a r l y as a f u n c t i o n o f A as i n c a s e ( i i i ) . T h e r e f o r e t h e most s t a b l e n u c l e a r c o n f i g u r a t i o n c o r r e s p o n d s t o an a l t e r n a t i n g s t r u c t u r e i n a l l t h r e e c a s e s . Polymethneimine i s d i f f e r e n t : here the d e n s i t y m a t r i x i s o f d i a g o n a l C D W - t y p e , and t h i s c o r r e s p o n d s t o c a s e ( i i ) . A c t u a l l y t h e a l t e r n a t i n g g e o m e t r y i s s l i g h t l y more f a v o r a b l e (1CJ ) , b u t t h i s c o n c l u s i o n may depend e . g . on t h e b a s i s s e t u s e d . The E n e r g y Gap I n T r a n s

(CH)^:

From the above r e s u l t s i t f o l l o w s t h a t t h e most stable g e o m e t r y i s an a l t e r n a t i n g one a t HF l e v e l w i t h n o n z e r o E . Its a c t u a l v a l u e , however, i s too l a r g e i n comparison w i t h the § p t i c a l a b s o r p t i o n maximum o b s e r v e d b e l o w 2 eV (9). This problem o f the t h e o r e t i c a l o v e r e s t i m a t i o n o f t h e e n e r g y gap i s a known d r a w b a c k o f t h e ab i n i t i o H a r t r e e - F o c k method ( 1 1 ) and s e r v e s as one o f t h e m a i n m o t i v a t i o n s t o go b e y o n d t h e HF d e s c r i p t i o n . Unfortunately, as we have r e c e n t l y o b s e r v e d ( 1 2 ) , t h e s i m p l e s t s u c h d e s c r i p t i o n ( t h e s p i n - u n r e s t r i c t e d HF method) does n o t l o w e r t h e gaps a t a l l for the polymers i n q u e s t i o n i n c o n t r a s t to the s u g g e s t i o n o f M i s u r k i n and O v c h i n n i k o v (13). The s i t u a t i o n i s c o m p l i c a t e d by the d i f f i c u l t y i n s e p a r a t i n g the e x c i t a t i o n s c r e a t i n g free e l e c t r o n s and h o l e s from t h e bound e l e c t r o n - h o l e quasi-particles (excitons). O n l y t h e former c o r r e s p o n d t o t h e E , w h i l e t h e c r e a t i o n o f the energy o f the l a t t e r , E , l i e s w i f t i i n the g a p . E -E may be even a l a r g e f r a c t i o n o f £ , and t h e e x p e r i m e n t s wf?icfi may d i s t i n g u i s h b e t w e e n E and E ( ] § i o t o c o n d u c t i o n , p h o t o v o l t a i c e f f e c t ) a r e d i f f i c u l t §6* carry**out t o y i e l d unambiguous result. One o f u s h a s r e c e n t l y c a r r i e d o u t an e x c i t o n c a l c u l a t i o n u s i n g a P a r i s e r - P a r r - P o p l e (PPP) model (14) w o r k i n g out e x c i t o n f o r m a l i s m i n the i n t e r m e d i a t e b i n d i n g ( l o c a l i z e d o n , b u t a l s o d e l o c a l i z e d o v e r s e v e r a l (CH) u n i t s ) . A c c o r d i n g t o t h i s as w e l l as o t h e r e x c i t o n c a l c u l a t i o n s (15) > t h e 2 eV p e a k i n ( C H ) may v e r y w e l l be due t o e x c i t o n s . I n c a s e o f t h e p o l y d i a c e t y l e n e s t h e d i f f e r e n c e o f t h e p h o t o c o n d u c t i o n o n s e t and t h e a b s o r p t i o n t h r e s h o l d i s ~ 0 . 5 e V , t h e r e f o r e i n t h a t c a s e t h e e x c i t o n p i c t u r e seems well established (16). e

X

x

The B l u e S h i f t

In Polydiacetylenes:

The s t r u c t u r e o f some p o l y d i a c e t y l e n e (PDA) s i n g l e c r y s t a l p o l y m e r s w h i c h c l o s e l y r e s e m b l e (CH) a r e known due t o e x i s t e n c e o f large s i n g l e c r y s t a l samples. The two p o s s i b l e s t r u c t u r e s o f

In Photon, Electron, and Ion Probes of Polymer Structure and Properties; Dwight, David W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

PHOTON, ELECTRON, AND ION PROBES

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 4, 2016 | http://pubs.acs.org Publication Date: August 13, 1981 | doi: 10.1021/bk-1981-0162.ch009

108

t h e m a i n c h a i n a r e t h e a c e t y l e n i c - t y p e , (= RC - C = C - RC=) , A T , and the butatriene-type - f R C = C = C = R C ) ^ , BT structures. T h e r e i s a s i g n i f i c a n t b l u e s h i f t i n the 2 eV r a n g e o p t i c a l a b s o r p t i o n s p e c t r u m o f t h e s e m a t e r i a l s o f the o r d e r o f 0 . 3 eV g o i n g from AT t o BT s t r u c t u r e . We b e l i e v e , t h a t t h i s phenomenon i s a c o n s e q u e n c e o f the i n t e r a c t i o n o f the e l e c t r o n s w i t h t h e s t a t i c l a t t i c e o f the n u c l e i i n the main c h a i n . I t s u n d e r s t a n d i n g i s d e s i r a b l e as t h e e x p e r i m e n t a l f a c t s a r e much b e t t e r known f o r DPA t h a n f o r (CH) , a l t h o u g h the a l t e r n a t i o n p r o b l e m h e r e i s p r i n c i p a l l y d i f f e r e n t from t h a t o f (CH) since both structures have a l r e a d y a n o n - z e r o g a p , and p r o b a b l y s i d e c h a i n s a r e f i x i n g t h e AT o r BT t y p e s t r u c t u r e . We have c a r r i e d o u t c r y s t a l o r b i t a l c a l c u l a t i o n s u s i n g t h e H u c k e l - m o d e l ( a s W i l s o n (11) d i d ) as w e l l as t h e more r e f i n e d s e l f - c o n s i s t e n t PPP m o d e l . N e x t t a b l e summarizes a few r e s u l t s ( i n eV). A s i s a p p a r e n t , t h e H u c k e l v a l u e s a r e s m a l l and a b i g red s h i f t is obtained. The PPP m o d e l w i t h u n i f o r m 3*s (but repulsion integrals t a k e n a c c o r d i n g to a l t e r n a t i n g geometry) p r e d i c t s b l u e s h i f t and E i n r o u g h agreement w i t h e x p e r i m e n t . T r u e PPP m o d e l w i t h a l t e r n a t i n g 3 ' s g i v e s v e r y l a r g e gaps and a small blue s h i f t . Table

Methods

Acetylene

I

Butatriene

Shift

Experiment

2.0

2.3

Huckel

1.487

0.578

1.843

2.129

+0.286

5.855

5.905

+0.050

PPP (all PPP a) b)

a

'

3=-2.39eV) ^

+0.3 b

)

-0.909

U s i n g v a l u e s d e p e n d i n g on bond l e n g t h a l a L . S a l e m ( 2 ) ; Wilson value (17).

These c o m p u t a t i o n a l f i n d i n g s may be r a t i o n a l i z e d by t a k i n g i n t o a c c o u n t t h e exchange t e r m i n P P P . W i t h i n t h e f i r s t n e i g h b o r a p p r o x i m a t i o n E = | 2 $ * - 3 * - 3 | (17), where 3 > = 3 - 1/2 P .Y • . The i increx r e f e r s t o the s i n g l e ( s ) , d o u b l e C