12 Band Structure Calculations and Their Relations to Photoelectron Spectroscopy J E A N - M A R I E A N D R É , J O S E P H D E L H A L L E , and JEAN J A C Q U E S P I R E A U X
Downloaded by FUDAN UNIV on March 28, 2017 | http://pubs.acs.org Publication Date: August 13, 1981 | doi: 10.1021/bk-1981-0162.ch012
Facultés Universitaires Notre-Dame de la Paix, 61, rue de Bruxelles, 5000-Namur, Belgium
M e t h o d s F o r The D e t e r m i n a t i o n o f O n e - E l e c t r o n Bands S i n c e t h e f i r s t t h e o r e t i c a l w o r k s o f t h e s i x t i e s (1_) on LCAO t e c h n i q u e s i n p o l y m e r quantum c h e m i s t r y , t h e f i e l d h a s known a r a p i d d e v e l o p m e n t and s t a n d a r d SCF c a l c u l a t i o n s on r e g u l a r p o l y mers a r e now r o u t i n e l y p e r f o r m e d . I n those methods, the t r a n s l a t i o n a l symmetry i s f u l l y e x p l o i t e d ( a n d c o n s e q u e n t l y assumed) i n o r d e r t o r e d u c e t o manageable d i m e n s i o n s t h e f o r m i d a b l e t a s k o f c o m p u t i n g e l e c t r o n i c s t a t e s o f an e x t e n d e d s y s t e m . I n the t h e o r e t i c a l d e s c r i p t i o n o f r e g u l a r p o l y m e r s , the m o n o e l e c t r o n i c l e v e l s ( o r b i t a l energies i n the m o l e c u l a r d e s c r i p t i o n ) a r e r e p r e s e n t e d as a m u l t i v a l u e d f u n c t i o n o f a r e c i p r o c a l wave number d e f i n e d i n t h e i n v e r s e s p a c e d i m e n s i o n . The s e t o f a l l those branches (energy bands) p l o t t e d v e r s u s the r e c i p r o c a l wave number ( k - p o i n t ) i n a w e l l d e f i n e d r e g i o n o f t h e r e c i p r o c a l s p a c e ( f i r s t B r i l l o u i n z o n e ) i s t h e band s t r u c t u r e o f t h e p o l y mers. I n t h e u s u a l t e r m i n o l o g y , we n o t e t h e a n a l o g y b e t w e e n t h e o c c u p i e d l e v e l s and t h e v a l e n c e b a n d s , t h e u n o c c u p i e d l e v e l s and the conduction band. The o n e - d i m e n s i o n a l p e r i o d i c l a t t i c e o f a p o l y m e r c o n s i s t s o f a l a r g e number, 2N + 1 (N-*») o f u n i t c e l l s ^ o f l e n g t h & e a c h containing n u c l e i at p o s i t i o n s % , ^ > •••> ^ ^ ^ c e l l s o r i g i n s j o z , j = 0 , ± 1 , ±2T, ±N, and 2 n electrons d i s t r i b u t e d a l o n g the c h a i n . A s t r i c t e l e c t r o n e u t r a ? i t y o f the cells (2n = + + . . . + Z^) i s necessary. The e l e c t r o n s a r e assumed t o d o u b l y o c c u p y a s e t o f o n e - e l e c t r o n o r b i t a l s , < $ > (k,r), o f B l o c h t y p e w r i t t e n as p e r i o d i c c o m b i n a t i o n s o f a) b a s i s f u n c tions, X (r). A
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