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Ab initio SCF MO study of H₆SI₂O₇ at simulated high pressure 1981

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AB INITIO SCF MO STUDY OF H 6 S I 2 0 7 AT SIMULATED HIGH PRESSURE B . S c i . , V i r g i n i a P o l y t e c h n i c I n s t i t u t e and S t a t e U n i v e r s i t y A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES Department of G e o l o g i c a l S c i e n c e s We accept t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August 1981 c Nancy Lee Ross, 1981 by MASTER OF SCIENCE i n In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Geological Sciences The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date August 6.,.19.81 DK-6 (2/79} ABSTRACT M o l e c u l a r o r b i t a l c a l c u l a t i o n s have been s u c c e s s i v e l y a p p l i e d t o m i n e r a l o g i c a l s t u d i e s of e q u i l i b r i u m m o l e c u l a r geometry, e l e c t r o n i c charge d i s t r i b u t i o n s , e l e c t r o n i c s p e c t r a and b u l k modulus c a l c u l a t i o n s . To d a t e , t h e s e s t u d i e s have m o d e l l e d bonding a t a t m o s p h e r i c p r e s s u r e . W i t h the ever i n c r e a s i n g i n t e r e s t i n h i g h p r e s s u r e phases and mantle m i n e r a l o g y , bonding s t u d i e s of m o l e c u l a r groups a t s i m u l a t e d h i g h p r e s s u r e can be an i n v a l u a b l e a i d t o u n d e r s t a n d i n g h i g h p r e s s u r e c r y s t a l c h e m i s t r y , bond e n e r g e t i c s and e l e c t r o n i c s p e c t r a . T h i s i n v e s t i g a t i o n t e s t s the f e a s i b i l i t y of v a r i o u s models t o s i m u l a t e p r e s s u r e i n ab i n i t i o SCF MO c a l c u l a t i o n s on common metal-oxygen p o l y h e d r a . P r e s s u r e i s s i m u l a t e d i n the c l u s t e r , H 6 S i 2 0 7 , by s y s t e m a t i c a l l y s t e p p i n g h e l i u m atoms d i r e c t e d ^ a l o n g the S i - 0 b r i d g i n g v e c t o r s toward the b r i d g i n g oxygen. Changes i n the S i - 0 bond l e n g t h s , S i O S i a n g l e s and S i - 0 f o r c e c o n s t a n t s are m o n i t o r e d w i t h i n c r e a s i n g p r e s s u r e . For an i n c r e a s e of 60 kbar p r e s s u r e , the S i - 0 bond l e n g t h and S i O S i a n g l e d e c r e a s e 0.30% and 4.5%, r e s p e c t i v e l y , w hich compares w e l l w i t h the 0.30% and 6.6% d e c r e a s e o b s e r v e d i n c- q u a r t z f o r a s i m i l a r i n crement of p r e s s u r e . The l i n e a r c o r r e l a t i o n of S i - 0 bond l e n g t h and - s e c ( S i O S i ) , known t o o c c u r a t one b a r , h o l d s a t e l e v a t e d p r e s s u r e . In a d d i t i o n , the S i - 0 s t r e t c h i n g and S i O S i b e n d i n g f o r c e c o n s t a n t s show a p e r c e n t a g e i n c r e a s e i n the r a t i o 1:6 up t o an e s t i m a t e d p r e s s u r e o f 140 kbar. iv TABLE OF CONTENTS Page ABSTRACT i i . LIST OF TABLES . .... v. LIST OF FIGURES v i . ACKNOWLEDGEMENTS , v i i i . I . INTRODUCTION 1 I I . MOLECULAR ORBITAL METHOD 5 D e s c r i p t i o n 5 MO Methods 11 I I I . CALCULATIONS 14 IV. MODELS 17 V. RESULTS AND "DISCUSSION 22 Model I 22 Model I I 25 V I . CONCLUSIONS 46 REFERENCES 49 V LIST OF TABLES Tab l e Page I . Asymmetric s t r e t c h i n g f o r c e c o n s t a n t s ( k a ) c a l c u l a t e d a t 1 bar f o r the c l u s t e r s H 6 S i 2 0 7 , H ? A 1 ? 0 7 - 2 , H 1 2 S i 5 0 , And H, 2 A l ' S i , 0 , " 1 w i t h a l l S i O S i and A l O S i a n g l e s e q u a l t o 180°. 24 I I . Comparison a t 1 bar of c a l c u l a t e d symmetric s t r e t c h , v s , asymmetric s t r e t c h , va , and b e n d i n g , , f r e q u e n c i e s f o r H 6 S i 2 0 7 w i t h those d e t e r m i n e d from i n f r a r e d and raman s p e c t r a f o r S i 2 0 7 " 6 , ( 0 ( S i ( C H 3 ) 2 ) a , and B a T i O S i 2 0 7 . 28 I I I . M u l l i k e n bond o v e r l a p p o p u l a t i o n s , n(Si-Ob) and n ( S i . . . S i ) , and atomic charges on b r i d g i n g oxygen, Q(Ob), and s i l i c o n , Q ( S i ) , f o r H 6 S i 2 0 7 a t 1 b a r , . 60 kbar and 140 kbar; the b r i d g i n g S i - 0 bond and S i O S i a n g l e a r e o p t i m i z e d . 38 vi LIST OF FIGURES F i g u r e Page 1. M o l e c u l a r c o n f o r m a t i o n f o r the dimers s t u d i e d w i t h model I (note the s t r a i g h t b r i d g i n g a n g l e ) ; p r e s s u r e i s s i m u l a t e d by d e c r e a s i n g the i n t e r t e t r a h e d r a l d i s t a n c e . 19 2. M o l e c u l a r c o n f o r m a t i o n f o r H 6 S i 2 0 7 s t u d i e d w i t h model I I . Note the bent b r i d g i n g a n g l e and p o s i t i o n i n g of h e l i u m atoms used t o s i m u l a t e p r e s s u r e by s y s t e m a t i c a l l y d e c r e a s i n g the d(He- Ojj) d i s t a n c e s . 20 3. Log of the asymmetric S i - 0 s t r e t c h i n g f o r c e c o n s t a n t , l o g ( k a ) , p l o t t e d a g a i n s t the l o g of the i n t e r t e t r a h e d r a l d i s t a n c e , l o g ( d ( T . . . T ) ) , f o r H 6 S i 2 0 7 where log(k*» ) =-7.351og(Si..Si) +4.55 ( r 2 = 0.999) H 6 A 1 2 0 7 - 2 : l o g ( k a ) =-7.421og(Al...Al) +4.63 (r 2=0.998) H , 2 S i 5 0 a : l o g ( k a ) =-7.121og(Si...Si) +4.47 (r 2=0.999) and H , 2 A l S i flO«: l o g ( k a ) = - 7 . 4 1 l o g ( A l . . . S i ) + 4 . 6 2 ( r 2 = 0 . 9 9 8 ) . 26 4. A comparison of asymmetric s t r e t c h i n g f r e q u e n c y , va , p l o t t e d a g a i n s t b r i d g i n g bond l e n g t h , d ( S i - O u ), f o r a group of t w e l v e p y r o s i l i c a t e s (a) and H 6 S i 2 0 7 ( b ) ; vQ's were d e t e r m i n e d from s p e c t r o s c o p i c e x periments f o r p y r o s i l i c a t e s whereas i / a ' s f o r H 6 S i 2 0 7 were c a l c u l a t e d . 29 5. The p o t e n t i a l energy s u r f a c e s f o r H 6 S i 2 0 7 and H e 2 H 6 S i 2 0 7 at 1 bar and 140 kbar, r e s p e c t i v e l y , p l o t t e d as a f u n c t i o n of the b r i d g i n g bond l e n g t h , d(Si-Ofc,), and the S i O S i a n g l e . 31 6. A comparison of the p o t e n t i a l energy c u r v e s f o r H 6 S i 2 0 7 and H e 2 H 6 S i 2 0 7 p l o t t e d as a f u n c t i o n of the b r i d g i n g d i s t a n c e , d ( S i - O b ), at 1 bar (upper c u r v e ) and 140 kbar (lower c u r v e ) , r e s p e c t i v e l y . 33 7. A comparison of the p o t e n t i a l energy c u r v e s f o r H 6 S i 2 0 7 and H e 2 H 6 S i 2 0 7 p l o t t e d as a f u n c t i o n of the S i O S i a n g l e a t 1 bar (upper curve) and 140 kbar (lower c u r v e ) , r e s p e c t i v e l y . 34 Symmetric, s t r e t c h i n g S i - 0 f o r c e c o n s t a n t , k 5, p l o t t e d a g a i n s t the S i O S i a n g l e a t 1 bar ( l e f t ) where k s =0.038(SiOSi ) + 1 .941 , r 2 = 0.97, and 140 kbar ( r i g h t ) where kg =0.040(SiOSi)+3.964, r 2=0.93. M u l l i k e n bond o v e r l a p p o p u l a t i o n , , n ( S i - 0 ^ ), p l o t t e d a g a i n s t the b r i d g i n g S i - 0 d i s t a n c e a t 1 bar (a) and a g a i n s t the symmetric s t r e t c h i n g f o r c e c o n s t a n t a t 1 bar (b) w i t h r 2 v a l u e s of 0.997 and 0.989 , r e s p e c t i v e l y ; the c o r r e s p o n d i n g r e l a t i o n s h i p s a t 140 kbar*" are found i n (c) and (d) w i t h r 2 v a l u e s of 0.999 and 0.971, r e s p e c t i v e l y . M u l l i k e n bond o v e r l a p p o p u l a t i o n , n ( S i - O b ' ) , p l o t t e d a g a i n s t the b r i d g i n g S i O S i a n g l e a t 1 bar (a) and a g a i n s t the p e r c e n t a g e s - c h a r a c t e r of the h y b r i d o r b i t a l s on the b r i d g i n g oxygen, 1 0 0 / ( 1 2 ) , a t 1 bar (b) w i t h the c o r r e s p o n d i n g r e l a t i o n s h i p s a t 140 kbar found i n (c) and ( d ) . The c u r v i l i n e a r t r e n d s of (a) and (c) both become l i n e a r i n (b) and ( d ) . The r e l a t i o n s h i p between the b r i d g i n g S i - 0 d i s t a n c e and - s e c ( S i O S i ) f o r H 6 S i 2 0 7 a t 1 bar and an e l e v a t e d p r e s s u r e e s t i m a t e d t o be 140 kbar. A comparison between the average S i - 0 b r i d g i n g d i s t a n c e p l o t t e d a g a i n s t - s e c ( S i O S i ) f o r c o e s i t e ( l e f t ) and H 6 S i 2 0 7 ( r i g h t ) ; a t 1 bar and 52 kbar, the r 2 v a l u e s f o r c o e s i t e based on the e x p e r i m e n t a l d a t a of L e v i e n and P r e w i t t (1981) are 0.97 and 0.90, r e s p e c t i v e l y ; the r 2 v a l u e s based on c a l c u l a t i o n s a t 1 bar and 60 kbar f o r H 6 S i 2 0 7 a r e 0.97 and 0.98, r e s p e c t i v e l y . I l l u s t r a t i o n of how e s t i m a t e s of k ^ x r o u g h l y e q u a l t o 60 kbar p r e s s u r e were o b t a i n e d . M o d e l l i n g changes t h a t o c c u r i n c - q u a r t z a t t h i s p r e s s u r e , d(Si-Ob) was kept c o n s t a n t w h i l e d e c r e a s i n g the S i O S i a n g l e from 144° t o 134° (path A-C); p a t h B-C shows the Ax a s s o c i a t e d w i t h an increment of 60 kbar p r e s s u r e . ACKNOWLEDGEMENTS S i n c e r e thanks a re extended t o Dr. E.P. Meagher f o r h i s gu i d a n c e , support and encouragement throughout t h i s s t u d y . T h i s work was s u p p o r t e d by the N a t i o n a l S c i e n c e and E n g i n e e r i n g Research C o u n c i l w i t h NSERC g r a n t 67-7061 and summer g r a n t s were p r o v i d e d by the NAHS. The c o o p e r a t i o n of the computing c e n t r e a t the U n i v e r s i t y of B r i t i s h Columbia i s a l s o g r a t e f u l l y acknowledged. A p p r e c i a t i o n i s e x p r e s s e d t o Dr. G.V. Gibbs f o r i n t r o d u c i n g me t o the e x c i t i n g w o r l d of m o l e c u l a r o r b i t a l t h e o r y and t o Monique Roussy f o r her many f r u i t f u l d i s c u s s i o n s . F i n a l l y I thank Gord Hodge f o r h i s d e f t hand a t d r a u g h t i n g and h i s concern f o r the a e s t h e t i c a p p e a l of a l l i l l u s t r a t i o n s used i n the t e x t . I . INTRODUCTION S i g n i f i c a n t advances have been made i n the p a s t twenty f i v e y e a r s w i t h r e g a r d s t o the a c c u r a t e d e t e r m i n a t i o n of s i l i c a t e s t r u c t u r e s which have, i n t u r n , s u p p l i e d a w e a l t h of da t a f o r c r y s t a l c h e m i c a l i n v e s t i g a t i o n s of t h i s g e o l o g i c a l l y i m p o r t a n t m i n e r a l group. For the most p a r t , these i n v e s t i g a t i o n s have d e a l t w i t h s t r u c t u r a l v a r i a t i o n s as a f u n c t i o n of s u b s t i t u e n t c a t i o n r a d i u s , t e m p e r a t u r e , and i n r e c e n t y e a r s , p r e s s u r e ( P a p i k e e t a l . , 1969; Cameron et. a l . , 1973; L e v i e n and P r e w i t t , 1981). U n t i l r e c e n t l y , i n v e s t i g a t i o n s d e a l i n g w i t h the c h e m i c a l bonding i n s i l i c a t e m i n e r a l s have been few i n number and have been based m a i n l y on the e l e c t r o s t a t i c model ( W h i t t a k e r , 1971; Ohashi and Burnham, 1972). W i t h the g e n e r a l knowledge t h a t s i l i c a t e s have a h i g h c o v a l e n t c h a r a c t e r i n t h e i r c h e m i c a l bonding ( P a u l i n g , 1981), t h e r e has been a t r e n d i n the p a s t decade toward u t i l i z i n g m o l e c u l a r o r b i t a l methods i n s i l i c a t e bonding s t u d i e s . In p a r t i c u l a r , t h e r e has been a c o n c e r t e d e f f o r t t o u n d e r s t a n d the s t e r e o c h e m i s t r y of s i l i c a t e s u s i n g m o l e c u l a r o r b i t a l f o r m a l i s m s r a n g i n g from the s e m i - e m p i r i c a l extended H u c k e l method ( L o u i s n a t h a n and G i b b s , 1972) and the CNDO/2 method (Meagher e_t a l . , 1979) t o the more s o p h i s t i c a t e d s e l f - c o n s i s t e n t f i e l d (SCF) ab i n i t i o method (Newton and G i b b s , 1980). In a d d i t i o n t o the s u c c e s s of the m o l e c u l a r o r b i t a l method i n s t e r e o c h e m i c a l s t u d i e s , i t has a l s o been a p p l i e d s u c c e s s i v e l y t o b u l k modulus c a l c u l a t i o n s (Newton e_t a l . , 1980) and t o the i n t e r p r e t a t i o n of a b s o r p t i o n , e m i s s i o n and p h o t o e l e c t r o n i c s p e c t r a i n s i l i c a and s i l i c a t e m i n e r a l s ( T o s s e l l , 1973, 1979; Dejong and Brown, 1980). The agreement between m o l e c u l a r o r b i t a l c a l c u l a t i o n s and observed v a l u e s f o r s i l i c a t e s s u p p o r t s the view t h a t i s o l a t e d m o l e c u l a r groups p o s s e s s l o c a l bonding f o r c e s t h a t are s i m i l a r t o those found i n t h r e e d i m e n s i o n a l s o l i d s . To d a t e , t h e s e s t u d i e s have m o d e l l e d bonding a t a t m o s p h e r i c p r e s s u r e and m o l e c u l a r o r b i t a l c a l c u l a t i o n s have n o t , as a r u l e , been a p p l i e d t o thermodynamic p r o p e r t i e s of m i n e r a l s . The q u a n t i t i e s K ( b u l k modulus) and dK/dP ( f i r s t d e r i v a t i v e of the b u l k modulus w i t h r e s p e c t t o p r e s s u r e ) a r e i m p o r t a n t parameters i n the e q u a t i o n s of s t a t e employed i n g e o p h y s i c a l r e s e a r c h and i n h i g h p r e s s u r e c r y s t a l c h e m i c a l s t u d i e s of m i n e r a l s . U n f o r t u n a t e l y t h e s e q u a n t i t i e s are d i f f i c u l t t o determine e x p e r i m e n t a l l y , e s p e c i a l l y a t h i g h c o n f i n i n g p r e s s u r e s . Recent advances i n c r y s t a l s t r u c t u r e d e t e r m i n a t i o n s a t h i g h p r e s s u r e s by x-ray d i f f r a c t i o n methods have y i e l d e d some v a l u a b l e d a t a . The e x p e r i m e n t s are c u r r e n t l y l i m i t e d , however, t o a p p r o x i m a t e l y 60 kbars p r e s s u r e and f o r e s e e a b l e advances w i l l e x tend the p r e s s u r e range t o 200 k b a r s a t b e s t . Over the past f i f t y y e a r s , v a r i o u s e m p i r i c a l r e l a t i o n s h i p s between K and molar volumes of s o l i d s have been proposed. R e c e n t l y i n v e s t i g a t o r s have proposed an e m p i r i c a l r e l a t i o n s h i p between the b u l k modulus of - c a t i o n - a n i o n p o l y h e d r a (Kp) and the mean c a t i o n - a n i o n d i s t a n c e s a t one atmosphere p r e s s u r e (Hazen and F i n g e r , 1979). They suggest t h a t i n o r d e r t o p r e d i c t K of a complex s o l i d one must know the Kp v a l u e s of the component p o l y h e d r a i n the s o l i d . A l t h o u g h these r e l a t i o n s h i p s l e n d themselves t o p r e d i c t i n g c o m p r e s s i b i l i t i e s of s i m p l e s o l i d s a t low c o n f i n i n g p r e s s u r e s , they are not s u c c e s s f u l f o r more complex s o l i d s or f o r p r e d i c t i o n s of K a t h i g h c o n f i n i n g p r e s s u r e s . An a l t e r n a t i v e approach i s proposed whereby the q u a n t i t i e s Kp and d(Kp)/dP w i l l be computed u t i l i z i n g the r e l a t i o n s h i p , Kp = V 0 2 E / o r 2 ) ( d r / d V ) 2 = V ( k s ( d r / d V ) 2 (1) where V i s the volume of the p o l y h e d r o n , r i s the c a t i o n - a n i o n d i s t a n c e , E i s the t o t a l energy and ks i s the s t r e t c h i n g f o r c e c o n s t a n t . T h i s study i s the f i r s t i n a s e r i e s i n v e s t i g a t i n g c o m p r e s s i b i l i t i e s of the more common m e t a l - o x i d e p o l y h e d r a found i n the e a r t h ' s c r u s t and mantle. The groundwork f o r f u t u r e s t u d i e s i s l a i d by t e s t i n g models f o r s i m u l a t i o n of p r e s s u r e w i t h SCF m o l e c u l a r o r b i t a l c a l c u l a t i o n s . Among the m o l e c u l a r c l u s t e r s of g e o l o g i c a l i n t e r e s t i s the S i 2 0 7 dimer. In t h i s s t u d y , we m o n i t o r changes i n the s t e r e o c h e m i s t r y of H 6 S i 2 0 7 as a f u n c t i o n of p r e s s u r e as w e l l as changes i n the s t r e t c h i n g and bending f o r c e c o n s t a n t s of the S i O S i l i n k a g e w i t h p r e s s u r e . The com p u t a t i o n of p o l y h e d r a l b u l k moduli and t h e i r v a r i a t i o n w i t h p r e s s u r e w i l l be completed i n work now underway on the SiO« and AlO„ t e t r a h e d r a and i n f u t u r e work on o c t a h e d r a l o x y a n i o n c l u s t e r s of magnesium, aluminum and s i l i c o n . I n v e s t i g a t i o n s such as the above p r o v i d e i n s i g h t s i n t o the atomic responses t o p r e s s u r e i n s i l i c a t e s t r u c t u r e s . 5 I I . MOLECULAR ORBITAL METHOD D e s c r i p t i o n The m o l e c u l a r o r b i t a l (MO) method forms the u n d e r l y i n g b a s i s f o r the c a l c u l a t i o n s i n t h i s s t u d y . The MO method p r o v i d e s an approximate s o l u t i o n t o the S c h r b d i n g e r wave e q u a t i o n , f o r a m a n y - e l e c t r o n m o l e c u l e or c l u s t e r of atoms. T h i s i s e q u i v a l e n t t o an e i g e n v e c t o r (*) e i g e n v a l u e (E) problem. The c e n t r a l premise i n MO t h e o r y i s t h a t the complex many- e l e c t r o n w a v e f u n c t i o n , *, can be approxi m a t e d as an a n t i s y m m e t r i z e d product of o n e - e l e c t r o n w a v e f u n c t i o n s , c a l l e d m o l e c u l e r o r b i t a l s , where n i s the t o t a l number of e l e c t r o n s i n the system. The o p t i m a l w a v e f u n c t i o n , * ( a l s o known as the H a r t r e e - F o c k w a v e f u n c t i o n ) , w i l l be the one which m i n i m i z e s the t o t a l H*=E* (2) n (3) energy f o r an atomic c l u s t e r i n i t s ground s t a t e , 'E |, Ewo|= f**H*dr (4) where * i s the ma n y - e l e c t r o n w a v e f u n c t i o n d e f i n e d i n (3) and H i s the m a n y - e l e c t r o n H a m i l t o n i a n o p e r a t o r . I n c o r p o r a t e d i n the h a m i l t o n i a n . are the k i n e t i c and p o t e n t i a l e n e r g i e s of the n u c l e i and e l e c t r o n s i n the atomic group. I f the Born-Oppenheimer a p p r o x i m a t i o n i s a c c e p t e d , whereby the n u c l e i are c o n s i d e r e d f i x e d , the h a m i l t o n i a n f o r an atomic c l u s t e r w i t h m n u c l e i and i , j e l e c t r o n s can be ex p r e s s e d i n the f o l l o w i n g way, H = y ( - n V 2 M ) V 2 -V y ^ e V r . J XY(e 2/r,y) , ( 5 ) where X7\ i s the L a p l a c i a n o p e r a t o r . The f i r s t term r e p r e s e n t s the k i n e t i c energy of the e l e c t r o n s , the second term r e p r e s e n t s t h e i r p o t e n t i a l e n e r g i e s due t o a t t r a c t i o n w i t h the n u c l e i and the t h i r d term r e p r e s e n t s the r e p u l s i o n between e l e c t r o n s . The h a m i l t o n i a n i s f r e q u e n t l y d i v i d e d i n t o o n e - e l e c t r o n terms, H , and t w o - e l e c t r o n terms, e 2/r;', such t h a t H = / H-+> / _ | e 2 / r ^ ) . ( 6 ) The energy r e l a t i n g t o the o n e - e l e c t r o n o p e r a t o r ( a l s o known as the c o r e h a m i l t o n i a n ) i s 7 Em B V*'* ( i ) H.-*m ( i ) d T«' ( 7 ) where E r e p r e s e n t s the sum of the k i n e t i c and p o t e n t i a l energy due t o an e l e c t r o n o c c u p y i n g o r b i t a l * m . A t y p i c a l t w o - e l e c t r o n term r e p r e s e n t i n g t h e r e p u l s i v e p o t e n t i a l energy between e l e c t r o n s i , j i s V •-^•m ( i )V. ( i ) ( e 2 / r i J - ) V ( 3>*r» ( 3 > d r« d r J " f ( < ( i ) * „ ( i ) (eVr^.) < ( j ) * n ( j ) dr.drj- where J m n i s the Coulomb r e p u l s i v e energy and K m r ! i s the exchange energy. The t o t a l energy of the system can be ex p r e s s e d as 1 1 f o r m o l e c u l a r o r b i t a l s m and n. A f t e r d e f i n i n g the h a m i l t o n i a n , s u i t a b l e w a v e f u n c t i o n s , must be found which s a t i s f y the o n e - e l e c t r o n S c h r b d i n g e r e q u a t i o n , where the o p e r a t o r F i s the 'Hartree-Fock or e f f e c t i v e one- 8 e l e c t r o n H a m i l t o n i a n and i s the o n e - e l e c t r o n energy. In o t h e r words, t h e r e w i l l be a s e r i e s of which are e i g e n v e c t o r s of the l i n e a r o p e r a t o r F, each w i t h a unique energy e m . In p r a c t i c e the m o l e c u l a r o r b i t a l s , * ^ , a r e expanded i n terms of a c o n v e n i e n t b a s i s s e t of N atomic o r b i t a l s , #r, c e n t e r e d on the v a r i o u s atoms of the m o l e c u l e , N tn That i s , the m o l e c u l a r o r b i t a l s are e x p r e s s e d as a l i n e a r c o m b i n a t i o n of atomic o r b i t a l s (LCAO). The atomic o r b i t a l s can be any g e n e r a l s e t of s p e c i f i e d s i n g l e - e l e c t r o n f u n c t i o n s . The b e s t a p p r o x i m a t i o n s f o r the wavef u n c t i o n s , , w i l l be those t h a t g i v e the l o w e s t e n e r g i e s , t m . T h i s i s i n a ccordance w i t h the V a r i a t i o n P r i n c i p l e which s t a t e s t h a t the v a l u e of the c a l c u l a t e d energy i s always g r e a t e r than or e q u a l t o the t r u e ground s t a t e e l e c t r o n i c energy. The problem i s reduced t o f i n d i n g the s e t of c o e f f i c i e n t s , c r m , t h a t y i e l d s the l o w e s t energy. T h i s i s done by m i n i m i z i n g the energy w i t h r e s p e c t t o each of the c o e f f i c i e n t s . F o l l o w i n g t h i s method, the c o e f f i c i e n t s must s a t i s f y e q u a t i o n s which can be w r i t t e n i n m a t r i x form, FC m= € m S C m (12) where C m i s a column v e c t o r of MO c o e f f i c i e n t s , F i s the m a t r i x whose elements are d e f i n e d as 9 (13) where F=H+J-K and S i s the o v e r l a p m a t r i x w i t h e l e m e n t s , a r e s o l v e d i t e r a t i v e l y w i t h s u c c e s s i v e l y b e t t e r c r m and E v a l u e s u n t i l convergence ( s e l f - c o n s i s t e n c y ) i s a c h i e v e d . In a d d i t i o n t o the t o t a l m o l e c u l a r energy, we are i n t e r e s t e d i n the o r b i t a l p o p u l a t i o n a n a l y s i s which p a r t i t i o n s the t o t a l number of e l e c t r o n s i n the system i n t o v a r i o u s atomic and bond c o n t r i b u t i o n s ( M u l l i k e n , 1955). I n t e g r a t i o n of the t o t a l m o l e c u l a r o r b i t a l d e n s i t y f u n c t i o n (14) The s e c u l a r e q u a t i o n s (or Roothaan e q u a t i o n s ) , FC=SCE (15) (16) expanded i n terms of the atomic o r b i t a l b a s i s , N n (17) si r=i y i e l d s the t o t a l number of e l e c r o n s , n: n (18) The M u l l i k e n bond o v e r l a p p o p u l a t i o n f o r a p a i r of atoms, s - t , i s d e f i n e d by when summed over a l l atomic o r b i t a l s on c e n t e r s and a l l atomic o r b i t a l s on c e n t e r t . I f the o v e r l a p p o p u l a t i o n between two atoms i s p o s i t i v e , t hey a r e bonded; i f n e g a t i v e , they a r e a n t i b o n d e d . The atomic o r b i t a l p o p u l a t i o n f o r an atom s, q ( s ) , i s o b t a i n e d by summing the q u a n t i t y n ( s - t ) over a l l atomic o r b i t a l s on t : n (19) q ( s ) (20) The atomic charge of atom s, Q ( s ) , i s d e f i n e d by Q(s) = q 0 ( s ) - q ( s ) (21 ) where q 0 ( s ) i s the t o t a l number of e l e c t r o n s i n the ground s t a t e of the f r e e , n e u t r a l atom s. 11 MO Methods M o l e c u l a r o r b i t a l c a l c u l a t i o n s can be c l a s s i f i e d i n t o two g e n e r a l c a t e g o r i e s : "approximate m o l e c u l a r o r b i t a l methods" and " ab i n i t i o " c a l c u l a t i o n s . I n the approximate MO methods, a l a r g e p o r t i o n of the e l e c t r o n i n t e g r a l s i n v o l v e d i n the c a l c u l a t i o n a r e approx i m a t e d by known atomic q u a n t i t i e s and by the use of " s e m i - e m p i r i c a l " e x p r e s s i o n s f o r elements i n the Ha r t r e e - F o c k m a t r i x . The a p p r o x i m a t i o n s adopted f o r these i n t e g r a l s and the s e m i - e m p i r i c a l e x p r e s s i o n s a r e e v a l u a t e d w i t h r e s p e c t t o t h e i r a b i l i t y t o p r e d i c t e x p e r i m e n t a l r e s u l t s . One of t h e better-known approximate MO methods i s the Complete N e g l e c t of D i f f e r e n t i a l O v e r l a p (CNDO/2) method (P o p l e et a l . , 1965). As i t s name i m p l i e s , a l l e l e c t r o n r e p u l s i o n i n t e g r a l s of the " d i f f e r e n t i a l o v e r l a p " t y p e 1 a r e n e g l e c t e d . In a d d i t i o n , s e m i - e m p i r i c a l e x p r e s s i o n s a re used t o c a l c u l a t e the elements of the H a r t r e e - F o c k m a t r i x . CNDO/2 m o l e c u l a r o r b i t a l c a l c u l a t i o n s on d i s i l o x a n e ( T o s s e l l and G i b b s , 1977) and p y r o s i l i c i c a c i d (Meagher e t a l . ,1979) y i e l d minimum energy S i O S i a n g l e s i n c l o s e agreement w i t h observed v a l u e s f o r s i l i c a polymorphs and g l a s s . However, CNDO/2 c a l c u l a t i o n s tend t o d r a s t i c a l l y o v e r e s t i m a t e bond l e n g t h s f o r second row elements (Marsh and Gordon, 1976). 1An example of an e l e c r o n r e p u l s i o n i n t e g r a l of the d i f f e r e n t i a l o v e r l a p type i s J*j" *v( 1 ) 4>s( 1 ) (1 f^*^- (2) <t> v (2 )dr,dr z where # r , # 5 , 0 f , a n d * 0 a r e atomic o r b i t a l s . ~>~- 12 In r e c e n t y e a r s , we have seen the development of ab i n i t i o SCF MO c a l c u l a t i o n s and computer programs u s i n g G a u s s i a n e x p a n s i o n s of S l a t e r - t y p e o r b i t a l s . U n l i k e the approximate MO methods, ab i n i t i o c a l c u l a t i o n s attempt t o s o l v e the f u l l e l e c t r o n i c S c h r o d i n g e r equaton f o r a many- e l e c t r o n system. A f t e r d e f i n i n g the atomic p o s i t i o n s and wave f u n c t i o n s , a l l atomic o v e r l a p i n t e g r a l s , S r 5 , a r e c a l c u l a t e d . The k i n e t i c and p o t e n t i a l o n e - e l e c t r o n i n t e g r a l s which make up the c o r e h a m i l t o n i a n a re e v a l u a t e d n e x t . C a l c u l a t i o n of the t w o - e l e c t r o n i n t e g r a l s f o l l o w s . The use of G a u s s i a n - t y p e w a v e f u n c t i o n s f o r the atomic o r b i t a l s e x p e d i t e s the comp u t a t i o n of the s e i n t e g r a l s . An i n i t i a l guess of the H a r t r e e - F o c k m a t r i x i s made through a H u c k e l or extended H u c k e l a p p r o x i m a t i o n 2 or th r o u g h d i a g o n a l i z a t i o n of the core h a m i l t o n i a n . With the approx i m a t e d H a r t r e e - F o c k m a t r i x , the e i g e n v a l u e s (or m o l e c u l a r o r b i t a l e n e r g i e s , e m ) and e i g e n v e c t o r s ( c ^ ' s ) a r e s o l v e d . With s u c c e s s i v e l y b e t t e r c o e f f i c i e n t s and energy v a l u e s , the s e c u l a r e q u a t i o n s (15) are s o l v e d i t e r a t i v e l y u n t i l convergence i s a c h i e v e d . Ab i n i t i o c o mputations enable us t o s o l v e f o r e q u i l i b r i u m bond l e n g t h s and a n g l e s f o r m o l e c u l e s i n v o l v i n g f i r s t and second row elements w i t h a h i g h degree of a c c u r a c y ( C o l l i n s e t a l . , 1976). O p t i m i z e d T-0 d i s t a n c e s and TOT a n g l e s , f o r 2 W i t h the extended H u c k e l a p p r o x i m a t i o n , the elements of the H a r t r e e - F o c k m a t r i x a r e approx i m a t e d w i t h the V a l e n c e O r b i t a l I o n i z a t i o n . P o t e n t i a l (VOIP): F =VOIP(u) ; F =K(VOIP(u)+VOIP(v)) example, compare w e l l w i t h l o c a l g e o m e t r i e s i n s i l i c a polymorphs, s i l i c a t e s , and s i l o x a n e s (Meagher et a l . , 1979,'Newton and G i b b s , 1980). F u r t h e r m o r e , ab i n i t i o c a l c u l a t i o n s of q u a d r a t i c f o r c e c o n s t a n t s on a l a r g e number of p o l y a t o m i c m o l e c u l e s s a t i s f a c t o r i l y account f o r n e a r l y a l l e x p e r i m e n t a l t r e n d s (Newton e t a l . , 1970). For these r e a s o n s , ab i n i t i o c a l c u l a t i o n s were used i n t h i s s t u d y . I l l . CALCULATIONS Ab i n i t i o SCF m o l e c u l a r o r b i t a l c a l c u l a t i o n s were undertaken w i t h the G a u s s i a n 76 computer program ( B i n k l e y e_t a l . , 1978). . Throughout t h i s s t u d y , a m i n i m a l b a s i s s e t , <t>r, was adopted i n which each atomic o r b i t a l of the c o n s t i t u e n t atoms i s r e p r e s e n t e d by a s i n g l e S l a t e r - t y p e o r b i t a l (STO) b a s i s f u n c t i o n . For example, we are d e a l i n g w i t h n i n e STO b a s i s f u n c t i o n s f o r s i l i c o n and f i v e STO b a s i s f u n c t i o n s f o r oxygen. To ease the c o m p u t a t i o n of the t w o - e l e c t r o n i n t e g r a l s , the STO f u n c t i o n s are , i n t u r n , expanded as G a u s s i a n - t y p e o r b i t a l s (GTO's) (Hehre e t a l . , 1969). In the m i n i m a l b a s i s s e t c a l c u l a t i o n s used i n t h i s s t u d y ( r e f e r r e d t o as a m i n i m a l STO-3G b a s i s s e t ) , each STO i s r e p r e s e n t e d by a l i n e a r c o m b i n a t i o n of t h r e e G a u s s i a n f u n c t i o n s . Newton and G i b b s (1980) and Gibbs et a l . (1981) have shown t h a t a STO-3G mi n i m a l b a s i s s e t i s s u f f i c i e n t when s t u d y i n g the bond l e n g t h and a n g l e r e l a t i o n s h i p s f o r H 6 S i 2 0 7 . M o l e c u l a r o r b i t a l c a l c u l a t i o n s l e n d t h e m s e l v e s r e a d i l y t o the e v a l u a t i o n of f o r c e c o n s t a n t s (Newton e_t a l . , 1979). The p o t e n t i a l energy i s expanded i n terms of q, E = E 0 + (dE/dq)q + 0 . 5 0 2 E / 9 q 2 ) q 2 + • • • (22) which i s e i t h e r the d i s p l a c e m e n t from the e q u i l i b r i u m bond l e n g t h , r - r 0 , or a n g l e , 9-&0, depending on whether a s t r e t c h i n g f o r c e c o n s t a n t , k^, o r ' b e n d i n g f o r c e c o n s t a n t , kg, i s b e i n g c a l c u l a t e d . In t h i s s t u d y , r r e f e r s to the b r i d g i n g S i - 0 bond l e n g t h and 9 i s the S i O S i a n g l e ; r 0 and © 0 a r e t h e i r r e s p e c t i v e e q u i l i b r i u m v a l u e s . By d e f i n i t i o n , the q u a d r a t i c f o r c e c o n s t a n t i s t w i c e the c o e f f i c i e n t of the q u a d r a t i c term: k 5 = O zE/3q 2) Nm" 1 (23) k b = ( 3 2 E / 3 q 2 ) / r 2 Nm"1 (24) where q qnd r a r e d e f i n e d above. Thus the f o r c e c o n s t a n t s are found d i r e c t l y by f i t t i n g a p a r a b o l a t o the p o t e n t i a l energy c u r v e . Increments of 0.01 A about the e q u i l i b r i u m bond l e n g t h and 2° about the e q u i l i b r i u m b r i d g i n g a n g l e were used t o f i t the p a r a b o l a . With ranges of 0.05 A and 8° about the e q u i l i b r i u m bond l e n g t h and b r i d g i n g a n g l e , h i g h e r o r d e r terms i n the e x p a n s i o n of the p o t e n t i a l energy (22) were found t o be i n s i g n i f i c a n t . The d e f i n i t i o n of the bending f o r c e c o n s t a n t g i v e n above (24) i s p r e f e r r e d because i t y i e l d s the same dimen s i o n s ( f o r c e / l e n g t h ) as the s t r e t c h i n g f o r c e c o n s t a n t . Three p r i n c i p a l v i b r a t i o n a l f r e q u e n c i e s f o r the p y r o s i l i c i c a c i d m o l e c u l e can be d e t e r m i n e d from the S i - 0 s t r e t c h i n g and S i O S i bending f o r c e c o n s t a n t s by f o l l o w i n g the method o u t l i n e d by H e r z b e r g (1945) f o r a XY 2 m o l e c u l e . T r e a t i n g the c l u s t e r as an XY 2 m o l e c u l e , ( 0 ( H 3 S i 0 3 ) 2 ) , and assuming a v a l e n c e f o r c e f i e l d model , we can e x p r e s s the p o t e n t i a l energy as E' = 0 . 5 k s q r 2 + 0.5k sq© 2 (25) where qr i s the d i s p l a c e m e n t from the e q u i l i b r i u m bond l e n g t h and q© i s the d i s p l a c e m e n t from the e q u i l i b r i u m b r i d g i n g a n g l e . The v a l e n c e f o r c e model . assumes t h a t t h e r e a r e no c r o s s terms i n the p o t e n t i a l energy i f i t i s ex p r e s s e d i n terms of qr and q© . With the p o t e n t i a l energy d e f i n e d by ( 2 4 ) , we can d e r i v e the f o l l o w i n g e q u a t i o n s ( H e r z b e r g , 1945; p.169): 4 T T 2 I A 2 = (1 + (2m Y/m x) sin 2(© 0/2) ) k s / m y (26) 4 r r 2 (i/ g 2 + i / 2 ) = (1 + (2m Y/m x) cos 2(© 0/2) ) ks/m-^ + (1 + (2mT/m?<) sin 2(© 0/2) ) 2k &/m Y (27) 1 6 ^ " ^ ^ = 2(1 + (2m Y/m x) ) k sk s/m^ (28) where vs , v a and are the symmetric S i - 0 s t r e t c h i n g , a n t i s y m m e t r i c S i - 0 s t r e t c h i n g and S i O S i bending f r e q u e n c i e s , r e s p e c t i v e l y ; mK i s the mass of X ( 0 ) , m r i s the mass of Y ( H 3 O S i 3 ) and a l l o t h e r terms have been d e f i n e d p r e v i o u s l y . E q u a t i o n s (26),(27) and (28) are s o l v e d s i m u l t a n e o u s l y f o r v5 , i/g and v b . IV. MODELS B a s i c a l l y two d i f f e r e n t models were used i n an e f f o r t t o s i m u l a t e e l e v a t e d p r e s s u r e s i n our c a l c u l a t i o n s . In the i n i t i a l model, which we w i l l r e f e r t o as model I , p r e s s u r e was s i m u l a t e d by .simply l o c k i n g the S i . . . S i d i s t a n c e a t s u c c e s s i v e l y s h o r t e r v a l u e s w h i l e m a i n t a i n i n g a s t r a i g h t S i O S i a n g l e . T h i s model has r e s t r i c t e d a p p l i c a t i o n s because of the need t o m a i n t a i n a s t r a i g h t S i O S i a n g l e . However, model I was u s e f u l i n comparing symmetric and asymmetric s t r e t c h i n g f o r c e c o n s t a n t s a t one bar and asymmetric f o r c e c o n s t a n t s a t e l e v a t e d p r e s s u r e s f o r the c l u s t e r s H 6 S i 2 0 7 , H 6 A 1 2 0 7 ~ 2 , H 1 2 S i 5 0 ( , and H , 2 A1S i ,0," 1 . The symmetric f o r c e c o n s t a n t s were c a l c u l a t e d by keeping the b r i d g i n g oxygen immobile w h i l e m o n i t o r i n g the changes i n energy as the S i atoms were brought i n toward the oxygen. The asymmetric f o r c e c o n s t a n t s , on the o t h e r hand, were c a l c u l a t e d by m a i n t a i n i n g a c o n s t a n t S i . . . S i d i s t a n c e w h i l e m o n i t o r i n g the changes i n energy as the b r i d g i n g oxygen was o s c i l l a t e d . We a l s o used model I t o study the e f f e c t of p o l y m e r i z a t i o n on the S i - 0 f o r c e c o n s t a n t as w e l l as the e f f e c t t h a t s u b s t i t u t i n g aluminum f o r s i l i c o n has upon the s t r e t c h i n g f o r c e c o n s t a n t s a t one bar and as a f u n c t i o n of p r e s s u r e . In a l l of the c l u s t e r s s t u d i e d w i t h model I , s t a g g e r e d c o n f o r m a t i o n s were used and the S i O S i and A l O S i a n g l e s were m a i n t a i n e d a t 180° . In the H 6 S i 2 0 7 c l u s t e r , the 0-H bond l e n g t h s were 0.96 A w h i l e the SiOH and OSiO a n g l e s were l o c k e d a t 109.47° , r e s p e c t i v e l y ( F i g u r e 1). In the l a r g e r c l u s t e r s , the OSiH and OAlH a n g l e s were 109.47° w h i l e the S i - H d i s t a n c e s were l o c k e d a t 1.49 A. T e t r a h e d r a l , T<J , symmetry was m a i n t a i n e d w i t h i n the S i O a and AlO« t e t r a h e d r a throughout a l l co m p u t a t i o n s . In the second model, which we w i l l r e f e r t o as model I I , p r e s s u r e was s i m u l a t e d about an H 6 S i 2 0 7 c l u s t e r by p l a c i n g i n e r t h e l i u m atoms a l o n g the S i - 0 b r i d g i n g v e c t o r and s y s t e m a t i c a l l y s t e p p i n g the two h e l i u m s toward the b r i d g i n g oxygen. T h i s model a l l o w s f o r p r e s s u r e s i m u l a t i o n a t bent S i O S i a n g l e s and i s more p r e c i s e l y a u n i a x i a l s t r e s s d i r e c t e d a l o n g the S i - 0 v e c t o r s . The H 6 S i 2 0 7 dimer ( F i g u r e 2) was p l a c e d i n a s t a g g e r e d c o n f o r m a t i o n w i t h O-H d i s t a n c e s , d(O-H), and S i - 0 n o n b r i d g i n g bond l e n g t h s , d(Si-Ob ), of 0.96 A and 1.65 A, r e s p e c t i v e l y . The OSiO and SiOH a n g l e s were l i k e w i s e m a i n t a i n e d a t 109.47° and 180° , r e s p e c t i v e l y , throughout a l l c o m p u t a t i o n s . At one atmosphere, the e q u i l i b r i u m d i s t a n c e s were t h e same w i t h or w i t h o u t the h e l i u m atoms. At e l e v a t e d p r e s s u r e s , we found t h a t the M u l l i k e n bond o v e r l a p p o p u l a t i o n s between h e l i u m and n o n b r i d g i n g oxygens, n(He-Onb ), and h e l i u m and s i l i c o n , n ( H e - S i ) , were never g r e a t e r than 0.004 and 0.007, r e s p e c t i v e l y . Whereas model I y i e l d s asymmetric s t r e t c h i n g f o r c e c o n s t a n t s a t e l e v a t e d p r e s s u r e , model I I y i e l d s symmetric s t r e t c h i n g f o r c e c o n s t a n t s at one bar and a t p r e s s u r e . The Figure 1. Molecular conformation f o r the aimers studied with model I (note the s t r a i g h t bridging angle); pressure i s simulated by decreasing the i n t e r t e t r a h e d r a l distance. Figure 2. Malecular conformation for HgSi20y when studied with model I I . Note the bent bridging angle and positioning of helium atoms used to simulate pressure by systematically decreasing the dCHe-Ojj) distances. f o l l o w i n g were s t u d i e d w i t h model I I : 1) changes i n the e q u i l i b r i u m s t e r e o c h e m i s t r y of H 6 S i 2 0 7 as a f u n c t i o n of p r e s s u r e ; 2) changes i n the s t r e t c h i n g and bending f o r c e c o n s t a n t s w i t h p r e s s u r e ; and 3) the t o t a l p o t e n t i a l energy as a f u n c t i o n of b r i d g i n g bond l e n g t h s and a n g l e s at e l e v a t e d p r e s s u r e s . Model I I i s p r e f e r r e d because the b r i d g i n g a n g l e e n e r g e t i c s as w e l l as the b r i d g i n g bond e n e r g e t i c s can be s t u d i e d as p r e s s u r e i s i n c r e a s e d . 22 V. RESULTS AND DISCUSSION Model I As s t a t e d , model I p r o v i d e s a means of comparing the symmetric, ks , and asymmetric , k a , s t r e t c h i n g f o r c e c o n s t a n t s . At one b a r , the c a l c u l a t e d ks and k a f o r H 6 S i 2 0 7 3 a r e 774 Nm"1 and 861 Nm"1 , r e s p e c t i v e l y . S i m i l a r l y , the asymmetric s t r e t c h i n g f o r c e c o n s t a n t f o r H 6 A 1 2 0 7 " 2 * , 630 Nm"1 , i s lower than the symmetric s t r e t c h i n g f o r c e c o n s t a n t , 715 Nm"1 . These r e s u l t s c o n f l i c t w i t h c a l c u l a t i o n s based on i n f r a r e d and raman s p e c t r o s c o p i c d a t a f o r a S i 2 0 7 group w i t h a l i n e a r b r i d g e ( L a z a r e v , 1972). The c a l c u l a t e d asymmetric, i / . s , and symmetric, v s , s t r e t c h i n g f r e q u e n c i e s of the S i O S i b r i d g e i n d i c a t e the asymmetric f o r c e c o n s t a n t i s g r e a t e r . The asymmetric s t r e t c h i n g f r e q u e n c y i s e x p e c ted t o be h i g h e r s i n c e i t i n v o l v e s a l a r g e a m p l i t u d e of v i b r a t i o n f o r the l i g h t e r c e n t r a l atom and a s m a l l a m p l i t u d e of v i b r a t i o n f o r the t e r m i n a l groups. C o n v e r s e l y , vs s h o u l d be low s i n c e the c e n t r a l atom has a s m a l l a m p l i t u d e of v i b r a t i o n and the t e r m i n a l groups have a l a r g e a m p l i t u d e of v i b r a t i o n i n the 3The d(Si-O^) =1.65 A i n t h i s H 6 S i 2 0 7 c l u s t e r . "The d(Al-O^) =1.735 A i n t h i s H 6 A 1 2 0 7 " 2 c l u s t e r . 23 symmetric mode (Ross, 1972). Model I , however, p r e d i c t s the o p p o s i t e t o what i s e x p e c t e d . In a d d i t i o n t o comparing k s and k a , model I was used t o s t u d y t h e e f f e c t s t h a t p o l y m e r i z a t i o n and s u b s t i t u t i o n of A l f o r S i have upon . the f o r c e c o n s t a n t s a t one bar and'as a f u n c t i o n of p r e s s u r e . A comparison of asymmetric s t r e t c h i n g f o r c e c o n s t a n t s a t a t m o s p h e r i c p r e s s u r e f o r the d i m e r s , H 6 S i 2 0 7 and H 6 A 1 2 0 7 " 2 ,and h i g h l y - p o l y m e r i z e d c l u s t e r s , H 1 2 S i 5 0 « and H, 2 A l S i ,0,," 1 , i s found i n T a b l e I . With i n c r e a s i n g p o l y m e r i z a t i o n from H 6 S i 2 0 7 t o H ^ S i s O j , , the asymmetric s t r e t c h i n g f o r c e c o n s t a n t of d ( S i - 0 ] 0 ) does not i n c r e a s e s i g n i f i c a n t l y . S p e c t r o s c o p i c s t u d i e s on framework s i l i c a t e s show asymmetric s t r e t c h i n g f r e q u e n c i e s f o r TOT l i n k a g e s a r e i n the range 950-1200 cm" ' ( M i l k e y , 1960; Moenke, 1962; Lyon, 1962; Moenke, 1966). F u r t h e r m o r e , t h e s e v a l u e s o v e r l a p the range found f o r p y r o s i l i c a t e s and c h a i n s i l i c a t e s (Farmer, 1974) thus s u p p o r t i n g our r e s u l t s . A d e c r e a s e i n the asymmetric s t r e t c h i n g f o r c e c o n s t a n t from 788 Nm"1 t o 647 Nm"1 was found by s u b s t i t u t i n g aluminum f o r s i l i c o n i n the dimer. The c l u s t e r w i t h A l O S i l i n k a g e s , H , 2 A l S i " 1 , has an asymmetric s t r e t c h i n g f o r c e c o n s t a n t of 695 Nm"1 which i s l e s s than t h a t f o r the S i - 0 bond and g r e a t e r than t h a t f o r the A l - 0 bond showing t h a t t h e r e i s a g r a d u a l d e c r e a s e i n k $ as the aluminum c o n t e n t i n c r e a s e s . In k e e p i n g w i t h our r e s u l t s , M i l k e y (1960) has noted t h a t the c e n t e r of g r a v i t y of a b s o r p t i o n bands i n the r e g i o n 950-1200 cm" 1 tends t o s h i f t t o lower f r e q u e n c y w i t h i n c r e a s i n g T a b l e I . Asymmetric s t r e t c h i n g f o r c e c o n s t a n t s ( k B ) c a l c u l a t e d a t 1 bar f o r the c l u s t e r s H 6 S i 2 0 7 , H 6 A 1 2 0 7 - 2 , H 1 2 S i 5 0 „ , and H 1 2AlSi uO„" 1 w i t h a l l S i O S i and A l O S i a n g l e s e q u a l t o 180°. C l u s t e r kg,(Nm- 1 ) H 6 S i 2 0 7 788 H 6 A 1 2 0 7 - 2 647 H 1 2 S i 5 0 4 796 H, 2AlSi„O u- 1 695 25 aluminum c o n t e n t . C a l i b r a t i o n of p r e s s u r e f o r the c l u s t e r s s t u d i e d w i t h model I was not p o s s i b l e . We were, however, a b l e t o lo o k a t r e l a t i v e changes and v a l u e s of the f o r c e c o n s t a n t s w i t h i n c r e a s i n g p r e s s u r e by p l o t t i n g l o g ( k ) v e r s e s l o g ( d(T...T) where d(T...T) i s the i n t e r t e t r a h e d r a l d i s t a n c e ( F i g u r e 3 ) . As d(T...T) d e c r e a s e s the p r e s s u r e i n c r e a s e s , hence the p r e s s u r e i n c r e a s e s from r i g h t t o l e f t i n F i g u r e 3. The asymmetric s t r e t c h i n g f o r c e c o n s t a n t s f o r d ( A l - O b ) are c o n s i s t e n t l y lower than those f o r d(Si-Ofc ) w i t h i n c r e a s i n g p r e s s u r e . In a d d i t i o n , the f o r c e c o n s t a n t s f o r t h e . d i m e r s ( F i g u r e 3a) and the h i g h l y - p o l y m e r i z e d c l u s t e r s ( F i g u r e 3b) i n c r e a s e s i m i l a r l y w i t h d e c r e a s i n g i n t e r t e t r a h e d r a l d i s t a n c e s as seen by the n e a r l y p a r a l l e l t r e n d s . The use of model I v e r i f i e d t h e f e a s i b i l i t y of s t u d y i n g S i - 0 bond e n e r g e t i c s and f o r c e c o n s t a n t s a t s i m u l a t e d e l e v a t e d p r e s s u r e s w i t h ab i n i t i o SCF m o l e c u l a r o r b i t a l c a l c u l a t i o n s . The model was abandoned, however, i n f a v o r of model I I which a l l o w s us t o i n c o r p o r a t e the i m p o r t a n t s t r u c t u r a l v a r i a b l e of the S i O S i a n g l e . Model I I ' With the S i O S i bending f o r c e c o n s t a n t and symmetric S i - 0 s t r e t c h i n g f o r c e c o n s t a n t , we can s o l v e e q u a t i o n s ( 2 6 ) , (27) Figure 3. Log of the asymmetric Si-0 stretching force constant, l o g ( k a ) , plotted against the i n t e r - tetrahedral distance, log( d(T...T) ), for H 6 S i 2 0 7 where log(ka)=-7.351og(Si...Si)+4.55 (r2=0.999); H 6Al 20 7-2 where log(k a)=-7.421og(Al...Al)+4.63 (r2=0.998); Hi 2Si504 where log(k a)=-7.121og(Si...Si) +4.47 (r 2=0.999); and H 1 2A1SI4O4- 1 where log(k a)=-7.411og(Al...Si)+4.62 (r 2=0.998). 0.46 0.48 0.50 0.47 0.49 0.51 log (d (T • • • • T) ) l og (d (T • - T ) ) and (28) s i m u l t a n e o u s l y f o r vd ,. vs and vb• T a b l e I I p r e s e n t s a comparison between the v i b r a t i o n a l f r e q u e n c i e s c a l c u l a t e d f o r H 6 S i 2 0 7 a t one bar and t h o s e d etermined from i n f r a r e d and raman s p e c t r o s c o p i c e x p e r i m e n t s at one bar f o r compounds c o n t a i n i n g S i O S i l i n k a g e s . The S i 2 0 7 " 6 a n i o n , s i l o x a n e , ( 0 ( S i ( C H 3 ) 2 ) , , and p y r o s i l i c a t e , B a 2 T i O S i 2 0 7 , d i s p l a y a range of v a l u e s f o r the p r i n c i p a l v i b r a t i o n a l f r e q u e n c i e s . For example, i / 3 v a r i e s from 503-665 cm" 1 w h i l e va ranges from 1029-1104 cm" 1 . The o n l y bending v i b r a t i o n a l frequency a t t r i b u t e d s o l e l y t o S i O S i bending i s 169 cm" 1 f o r the S i 2 0 7 " 6 a n i o n . The c a l c u l a t e d v a l u e s show a r e a s o n a b l e agreement w i t h e x p e r i m e n t a l d a t a . The r e s u l t s are even more en c o u r a g i n g c o n s i d e r i n g we are comparing the e n e r g e t i c s of the S i O S i l i n k a g e i n H 6 S i 2 0 7 w i t h the e n e r g e t i c s of the S i O S i l i n k a g e i n v e r y complex compounds. T h i s l e n d s f u r t h e r support t o the premise t h a t the l o c a l bonding f o r c e s i n s i l o x a n e s and s i l i c a t e s a re s i m i l a r t o those i n i s o l a t e d m o l e c u l a r c l u s t e r s i n v o l v i n g the same atoms and c o o r d i n a t i o n number. In F i g u r e 4a, va i s p l o t t e d a g a i n s t the average b r i d g i n g S i - 0 bond l e n g t h f o r t w e l v e p y r o s i l i c a t e s a t a t m o s p h e r i c p r e s s u r e (Farmer, 1974). A s i m i l a r t r e n d i s found f o r H 6 S i 2 0 7 ( F i g u r e 4b) where the d i f f e r e n t d ( S i - 0 ^ ) c o r r e s p o n d t o c a l c u l a t e d e q u i l i b r i u m d i s t a n c e s a t d i f f e r e n t S i O S i a n g l e s . Both t r e n d s show a d e c r e a s e i n the asymmetric S i - 0 s t r e t c h i n g f r e q u e n c y as d ( S i - 0 b ) i n c r e a s e s . S i n c e v a i s d i r e c t l y p r o p o r t i o n a l t o the square r o o t of the symmetric s t r e t c h i n g f o r c e c o n s t a n t ( 1 9 ) , k s a l s o d e c r e a s e s as d(Si-Ob) i n c r e a s e s . T a b l e I I . Comparison a t 1 bar of c a l c u l a t e d symmetric s t r e t c h , vs, and asymmetric s t r e t c h , v-e,, and bending, i /b, f r e q u e n c i e s f o r H 6 S i 2 0 7 w i t h those determined from i n f r a r e d and raman s p e c t r a f o r S i 2 0 7 - 6 , (0(Si(CH 3) 2)„, and BaTiOSi 2 0 7 . C a l c u l a t e d F r e q u e n c i e s ( c m - 1 ) Exper i m e n t a l F r e q u e n c i e s ( c m - 1 ) H 6 S i 2 0 7 S i 2 0 7 0 ( S i ( C H 3 ) 2 ) , B a T i O S i 2 0 7 C 5 588 503 547 665 a 1 252 1 029 1 1 04 1039 b 1 33 1 69 G i l l e s p i e and Robinson, 1964. L a z a r e v , 1972. G a b e l i c a - R o b e r t and T a r t e , 1981. Figure 4. A comparison of the asymmetric stretching frequency,-v a, plotted against the bridging bond length, d(Si-0t>), for a group of twelve pyrosilicates (a) and H 6 S i 2 0 7 (b); v a's were determined from spectroscopic experiments for the pyrosilicates whereas v a ' s for H6Si207 were calculated. 1400 ,1400 1000 1300 1 1200H 1100 1.60 1000 1.68 1.56 H6Si207 (b) 1.60 1.64 d(Si-Ob) d(Si-Ob) 30 In o t h e r words, the b r i d g i n g S i - 0 bond becomes more i n c o m p r e s s i b l e ( t h a t i s , g r e a t e r k 5 ) as the b r i d g i n g bond l e n g t h d e c r e a s e s . The agreement between our c a l c u l a t i o n s and e x p e r i m e n t a l s t u d i e s a t a t m o s p h e r i c p r e s s u r e was e n c o u r a g i n g enough f o r us t o proceed w i t h the s i m u l a t i o n of p r e s s u r e . Because the e q u i l i b r i u m d(Si-Ob) d e c r e a s e s as the b r i d g i n g a n g l e widens at one bar (Newton and G i b b s , 1980), c o n s t a n t d(He-Ob) v a l u e s do not r e p r e s e n t e q u a l p r e s s u r e s at d i f f e r e n t b r i d g i n g a n g l e s . To a p p r o ximate e q u i v a l e n t p r e s s u r e s f o r d i f f e r e n t a n g u l a r c o n f i g u r a t i o n s , Hooke's Law was employed and the f a c t t h a t p r e s s u r e i s d i r e c t l y p r o p o r t i o n a l t o the f o r c e b e i n g a p p l i e d . T h e r e f o r e u n i t s of e q u i v a l e n t p r e s s u r e s e q u a l t o k-g,vAx were e s t a b l i s h e d where k 9 V Ax i s the average of the symmetric s t r e t c h i n g f o r c e c o n s t a n t over the i n t e r v a l , Ax, s t u d i e d . These p r o v i d e r e a s o n a b l e a p p r o x i m a t i o n s of e q u i v a l e n t p r e s s u r e s as l o n g as the i n t e r v a l , Ax, i s s m a l l . U s i n g t h i s method, a p o t e n t i a l energy s u r f a c e f o r H e 2 H 6 S i 2 0 7 was c o n s t r u c t e d as a f u n c t i o n of the b r i d g i n g S i - 0 bond l e n g t h and S i O S i a n g l e a t e l e v a t e d p r e s s u r e ( F i g u r e 5 ) . T h i s p r e s s u r e i s e s t i m a t e d t o be 140 kbar by methods e x p l a i n e d l a t e r . At one b a r , the energy s u r f a c e shows a l o n g , narrow v a l l e y s urrounded on t h r e e s i d e s by s t e e p energy b a r r i e r s ( F i g u r e 5 ) . The t o p o l o g y of the energy s u r f a c e changes n o t a b l y w i t h p r e s s u r e . At 140 kbar, the s u r f a c e shows a d i s t i n c t minimum surrounded on f o u r s i d e s by energy b a r r i e r s which a r e s i g n i f i c a n t l y s t e e p e r than those a t one b a r . Figure 5. Potential energy surfaces for HgSi20 7 at 1 bar and 140 kbar plotted as a function of the bridging distance, d(Si-Ob), and the SiOSi angle. ! /.SiOSi (deg) 32 Comparing the minimum of the energy t r o u g h a t one bar and. 140 kbar, we see t h a t t h e r e i s a n a r r o w i n g of the S i O S i a n g l e from 142° t o 132° and a d e c r e a s e of t h e b r i d g i n g S i - 0 bond from o o 1.585 A t o 1.565 A. The s t e e p e n i n g of the s i d e s of the energy s u r f a c e i s r e f l e c t e d by the i n c r e a s e i n k s from 743 Nm"1 a t one bar t o 913 Nm"1 a t 140 kbar ( F i g u r e 6) and an almost t r i p l i n g of k£ from 8.2 Nm"1 a t one bar t o 20.6 Nm"1 a t 140 kbar ( F i g u r e 7 ) . By t a k i n g v e r t i c a l c r o s s s e c t i o n s t h rough the p o t e n t i a l energy s u r f a c e s , the r e l a t i o n s h i p between ks and the S i O S i a n g l e can be s t u d i e d a t one bar and 140 kbar. F i g u r e 8 shows t h a t k s i n c r e a s e s as the b r i d g i n g a n g l e widens at the two p r e s s u r e s . E a r l i e r we i n v e s t i g a t e d the r e l a t i o n s h i p between and d(Si-Ob) a t a t m o s p h e r i c p r e s s u r e f o r H 6 S i 2 0 7 and a group of p y r o s i l i c a t e s ( F i g u r e 4) m e n t i o n i n g t h a t va i s d i r e c t l y p r o p o r t i o n a l t o k s . Newton and Gibbs (1980) have demonstrated a t one bar t h a t d ( S i - O ^ ) i s i n v e r s e l y c o r r e l a t e d w i t h the S i O S i a n g l e . T h e r e f o r e we a r e r e s t a t i n g the r e l a t i o n between va and d ( Si-O^) ( F i g u r e 4) i n terms of k s and the b r i d g i n g a n g l e ( F i g u r e 8 ) ; i n a d d i t i o n , we p r e d i c t t h a t t h i s r e l a t i o n s h i p h o l d s at p r e s s u r e . The i n c r e a s e i n ks and kg w i t h p r e s s u r e i s s u p p o r t e d by the i n f r a r e d s p e c t r o s c o p i c s t u d i e s of F e r r a r o and Manghnani (1972) and F e r r a r o e t a l . (1972) on c - q u a r t z and s i l i c a t e g l a s s e s at p r e s s u r e s up t o 58.8 kbar. They found t h a t the i n t e r t e t r a h e d r a l S i - 0 s t r e t c h i n g f r e q u e n c y f o r c - q u a r t z , f u s e d s i l i c a , V y c o r , and Pyrex • shows a p o s i t i v e dependence w i t h p r e s s u r e . The mixed OSiO and S i O S i bending f r e q u e n c y f o r c- 33 Figure 6. A comparison of the potential energy curves for HgSi20^ plotted as a function of the bridging distance, d(Si-O^), at 1 bar (upper curve) and 140 kbar (lower curve), where He2HgSi ? 0 7 i s the high p r e s s u r e phase. -1090.5170 .5171 .5172 .5173 H k s = 743 Nm d LU -1097.0968 H .0969 .0970 H .0971 .0972 1 4 k s = 912 Nm - 1 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 d(Si-Ob) (A) 34 Figure 7. A comparison of the p o t e n t i a l energy curves f o r E^S±20j p l o t t e d as a function of the SiOSi angle at 1 bar (upper curve; and 140 kbar (lower curve), where H e o H c S i o 0 7 i s the high pressure phase.. .. . b d / -1091.5172 .0974 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j 130.0 135.0 140.0 145.0 ASiOSi (deg) Figure 8. Symmetric Si-0 stretching force constant, k g, plotted against the SiOSi angle at 1 bar (left) where'ks»0.038(SiOSi)+l.941, r2=0.97, and 140 kbar (right) where kg=0.040(SiOSi)+3.964, r2=0.93. q u a r t z a l s o shows a p o s i t i v e dependence w i t h p r e s s u r e . The p r e s s u r e dependence noted f o r t h i s f r e q u e n c y p r i m a r i l y r e f l e c t s the change of the S i O S i a n g l e , l i n k i n g the t e t r a h e d r a . The r e s u l t s i n d i c a t e t h a t c o m p r e s s i o n of g l a s s t a k e s p l a c e a l o n g network c h a i n s c a u s i n g t e t r a h e d r a t o move c l o s e r t o one ano t h e r ( F e r r a r o et a l . , 1972). The change i n S i O S i a n g l e i s the most prominent e f f e c t of p r e s s u r e i n our c a l c u l a t i o n s , n a r r o w i n g 7.0% f o r a p r e s s u r e increment of 140 kbar w h i l e d ( S i - 0 ) d e c r e a s e s 1.3%. Recent h i g h - p r e s s u r e c r y s t a l l o g r a p h i c s t u d i e s of a- q u a r t z ( J o r g e n s e n , 1978; L e v i e n et. a l . , 1980) have a l s o shown t h a t the major e f f e c t of p r e s s u r e on the s t r u c t u r e i s t o c l o s e down the S i O S i a n g l e . Between one bar and 61.4 kbar, L e v i e n e t a l . (1980) found t h a t the average S i - 0 bond l e n g t h d e c r e a s e d 0.3% w h i l e the S i O S i a n g l e d e c r e a s e d 6.6%. In t h i s s t u d y , a c o m p a r a t i v e i n c r e a s e of 60 kbar r e s u l t e d i n a 0.3% decrease i n d(Si-Ofc ) and a 4.5% d e c r e a s e i n the b r i d g i n g a n g l e . Jorgensen (1978) and L e v i e n e_t a_l. (1980) performed the h i g h - p r e s s u r e e x p e r i m e n t s under h y d r o s t a t i c c o n d i t i o n s . With an i n c r e a s e i n p r e s s u r e , the framework of c o r n e r - l i n k e d t e t r a h e d r a can be c o l l a p s e d i d e a l l y ( t h e r e b y r e d u c i n g molar volume) by a c o o p e r a t i v e t i l t i n g of the r i g i d t e t r a h e d r a i n such a way t h a t the t e t r a h e d r a remain u n d i s t o r t e d ; the S i O S i a n g l e , however, i s reduced s i g n i f i c a n t l y . In our c a l c u l a t i o n s , a d i r e c t e d s t r e s s i s imposed by h e l i u m atoms p l a c e d a l o n g the S i - 0 v e c t o r s . The reason f o r t h i s i n t r i n s i c p r e f e r e n c e f o r a s m a l l e r S i O S i a n g l e w i t h i n c r e a s e d p r e s s u r e i n the H e 2 H 6 S i 2 0 7 m o l e c u l e i s not a p p a r e n t . The e l e c t r o n i c a d j u s t m e n t s w i t h i n c r e a s i n g p r e s s u r e a r e m i n i m a l as can be seen i n T a b l e I I I . There i s e s s e n t i a l l y no change i n the M u l l i k e n bond o v e r l a p p o p u l a t i o n n(Si-Ob) as w e l l as the net c h a r g e s on t h e b r i d g i n g oxygen and s i l i c o n s . L i k e w i s e g r o s s c harges on the v a l e n c e o r b i t a l s of s i l i c o n and oxygen show no s i g n i f i c a n t v a r i a t i o n . The i n c r e a s i n g n e g a t i v e v a l u e s of n ( S i . . . S i ) (Table I I I ) would tend t o f a v o r a wider S i O S i a n g l e w i t h i n c r e a s i n g p r e s s u r e . I t i s of i n t e r e s t , however, t h a t t h e m o l e c u l a r group shows an i n t r i n s i c p r e f e r e n c e f o r s m a l l e r S i O S i a n g l e s u n r e l a t e d t o volume c o n s i d e r a t i o n s . A l t h o u g h n(Si-Ob) e x h i b i t s no change w i t h i n c r e a s i n g p r e s s u r e ( T a b l e I I I ) , i t can be c o r r e l a t e d w i t h . d(Si-Ob) and k s when p r e s s u r e remains c o n s t a n t ( F i g u r e 9 ) . An i n c r e a s e of the e l e c t r o n i c o v e r l a p p o p u l a t i o n between S i and 0 r e s u l t s i n a s h o r t e r bond l e n g t h and a concommittant i n c r e a s e i n k£ a t one bar and 140 kbar. Newton and Gibbs (1980) have demonstrated a t one bar t h a t n ( S i - O i ; ) ) shows a c u r v i l i n e a r t r e n d when p l o t t e d a g a i n s t S i O S i but i s l i n e a r l y c o r r e l a t e d w i t h - s e c ( S i O S i ) . The l a t t e r c o r r e l a t i o n can be r e l a t e d t o h y b r i d i z a t i o n of the v a l e n c e o r b i t a l s on the b r i d g i n g oxygen of H 6 S i 2 0 7 (Brown et a l . , 1969). I f the h y b r i d o r b i t a l s on the oxygen a r e e x p r e s s e d i n the form s+X.p where X. i s the s-p m i x i n g c o e f f i c i e n t , i t can be shown t h a t X.2=-sec ( S i O S i ) ; f u r t h e r m o r e , the S i O S i a n g l e d e t e r m i n e s the p e r c e n t a g e s- c h a r a c t e r , 100/O+X. 2), of each h y b r i d (McWeeney, 1979). To i n v e s t i g a t e how p r e s s u r e a f f e c t s t h i s r e l a t i o n s h i p , nCSi-O^) T a b l e I I I . M u l l i k e n bond o v e r l a p p o p u l a t i o n s , n ( S i - 0^) and n ( S i . . . S i ) , and atomic charges on b r i d g i n g oxygen, Q ( 0 b ) , and s i l i c o n , Q ( S i ) , f o r H 6 S i 2 0 7 a t 1 bar , 6 0 kbar and 140 kbar; b r i d g i n g S i - 0 bonds and S i O S i a n g l e are o p t i m i z e d . (kbar) n(Si-O^) n ( S i . . . S i ) Q(O b) Q( S i 1x10" 3 +0.50 -0 .058 -0.70 1 .57 60 + 0.50 -0 .060 -0.70 1 .58 1 40 + 0.50 -o .062 -0.71 1 ...59 39 Figure 9. Mulliken bond overlap population, n(Si-O^), plotted against the bridging distance, d(Si-Ob), at 1 bar (a) and against the symmetric stretching force constant, ks, at 1 bar (b) with r 2 values of 0.997 and 0.989, respectively; the corresponding relationships at 140 kbar are found in (c) and (d) with r 2 values of 0.999 and 0.971, respectively. —1 1 1 1 1 I 1 I 1 1 1 1 1 1.55 1.57 1.59 900 1000 d ( S i - O b ) K s ( N m - 1 ) 40 v a l u e s were p l o t t e d a g a i n s t the b r i d g i n g a n g l e a t one bar ; and 140 kbar ( F i g u r e s 10a and 10c). The t r e n d s a t both p r e s s u r e s a r e c u r v i l i n e a r . On the o t h e r hand, when n ( S i - 0 ^ ) was p l o t t e d a g a i n s t the p e r c e n t a g e s - c h a r a c t e r of the b r i d g i n g oxygen a t the two p r e s s u r e s ( F i g u r e s 10b and I 0 d ) , w e l l - d e v e l o p e d l i n e a r c o r r e l a t i o n s (r 2=0.996 a t 1 b a r ; r 2=0.997 at 140 k bar) were o b t a i n e d . A c o r r e l a t i o n c l o s e l y r e l a t e d t $ the above i s the r e l a t i o n s h i p between d(Si-Ob) and - s e c ( S i O S i ) . At a t m o s p h e r i c p r e s s u r e , Newton and Gibbs (1980) have found t h a t a l i n e a r c o r r e l a t i o n e x i s t s between d ( S i - 0 | o ) and — s e c . ( S i O S i ) . With i n c r e a s i n g S i O S i , the s - c h a r a c t e r of the h y b r i d o r b i t a l s on the b r i d g i n g oxygen i n c r e a s e s and d ( S i - O b ) d e c r e a s e s . When observe d S i - 0 b r i d g i n g bond l e n g t h s i n c o e s i t e are p l o t t e d a g a i n s t - s e c ( S i O S i ) a t one bar (Gibbs e t a l . , 1977), a w e l l - d e v e l o p e d l i n e a r c o r r e l a t i o n (r 2=0.96) i s o b t a i n e d w i t h the s h o r t bonds i n v o l v i n g wide a n g l e s . I t has been suggested ( L e v i e n e t a l . , 1980; L e v i e n and P r e w i t t , 1981) t h a t t h i s r e l a t i o n s h i p f a i l s t o h o l d w i t h i n c r e a s i n g p r e s s u r e . However, one would not expect the r e l a t i o n t o h o l d f o r a g i v e n bond l e n g t h w i t h changing p r e s s u r e ; r a t h e r , one would expect the r e l a t i o n t o h o l d f o r a l l bond l e n g t h s i n a s t r u c t u r e a t c o n s t a n t p r e s s u r e whether i t be one bar or an e l e v a t e d p r e s s u r e . To i n v e s t i g a t e t h i s , we undertook a • study of the r e l a t i o n s h i p between d ( S i - O ^ ) and - s e c ( S i O S i ) a t an e l e v a t e d p r e s s u r e . F i g u r e 11 p r e s e n t s the r e s u l t s c o n f i r m i n g our p r e d i c t i o n s t h a t a s i g n i f i c a n t l i n e a r c o r r e l a t i o n e x i s t s a t a 4 1 F i gu re 10. M u l l i k e n bond o ve r l ap p o p u l a t i o n , n ( S i - 0 , ) , p l o t t e d a g a i n s t the b r i d g i n g S iOS i ang le a t 1 bar (a) and aga in s t the percentage s - c h a r a c t e r of the hyb r i d o r b i t a l s on the b r i d g i n g oxygen, 100/(1+ 2 ) , a t 1 bar (b) w i t h the cor respond ing r e l a t i o n s h i p s a t 140 kbar found i n (c) and (d ) . The c u r v i l i n e a r t rends i n (a) and (c) bo th become l i n e a r i n (b) and (d ) . Figure 11. The relationship between the bridging Si-0 distance and -sec(SiOSi) for H,Si„0-, at 1 bar and an elevated pressure estimated to be 140 kbar. 6 2 7 1.7 o< 1.5 # 1 bar ( r 2 - 0 . 9 7 ) ® Elev. Pressure ( r 2 s = 0 . 9 6 ) 1 — ; 1 1 1 1 1 1 — 2.0 1.9 : 1.8 1.7 1.6 1.5 1.4 1.3 1.1 1.0 -secZ_SiOSi no 4 3 g i v e n h i g h p r e s s u r e (r 2=0.96) as w e l l as 1 bar ( r 2 = 0 . 9 7 ) . Recent work on the s t r u c t u r e and c o m p r e s s i b i l i t y of c o e s i t e a t h i g h p r e s s u r e ( L e v i e n and P r e w i t t , 1981).supports t h i s f i n d i n g . When the average S i - 0 b r i d g i n g bond l e n g t h s a r e p l o t t e d a g a i n s t - s e c ( S i O S i ) a t 51.9 kbar, a s i g n i f i c a n t l i n e a r c o r r e l a t i o n (r 2=0.90) i s found. F i g u r e 12 compares the data f o r c o e s i t e a t one bar and. 51.9 kbar w i t h the c a l c u l a t e d d a t a f o r H 6 S i 2 0 7 a t one bar and 60 kb a r . The agreement between experiment and t h e o r y i s e n c o u r a g i n g . E s t i m a t e s of p r e s s u r e c o r r e s p o n d i n g t o kg^Ax terms were o b t a i n e d by m o d e l l i n g changes t h a t occur i n c - q u a r t z w i t h p r e s s u r e . L e v i e n e t a l . (1980) have noted a v e r y s l i g h t d e c r e a s e i n the mean S i - 0 d i s t a n c e and a s h i f t i n the S i O S i a n g l e from 143.7° t o 134.2° f o r an i n c r e a s e of 61.4 kbar p r e s s u r e . The k a v A x v a l u e c o r r e s p o n d i n g t o 61.4 kbar was ap p r o x i m a t e d by keeping d(Si-0|o ) c o n s t a n t i n H 6 S i 2 0 7 w h i l e d e c r e a s i n g S i O S i from 144° t o 134°. D i a g r a m m a t i c a l l y t h i s i s p a t h A-C i n F i g u r e 13. P a t h B-C shows t h a t t h e r e i s a s i g n i f i c a n t Ax a s s o c i a t e d w i t h a change i n p r e s s u r e of 61.4 kbar. The v a l u e of 140 kbar f o r k a y A x used i n many of the p r e c e d i n g c a l c u l a t i o n s was e s t i m a t e d by e x t r a p o l a t i o n from the 61.4 kbar v a l u e . Figure 12. A comparison between the average Si-0 bridging distance plotted against -sec(SiOSi) for coesite ( l e f t ) and H 6 S i 2 0 7 ( r i g h t ) ; at 1 bar and 52 kbar, the r 2 values for coesite based on experimental data from Levien and Prewitt (1981) are 0.97 and 0.90, respectively; the r 2 values based on calculations at 1 bar and 60 kbar for HgSi207 are 0.97 and 0.98, respectively. Figure 13. I l l u s t r a t i o n of how estimates of k a V A x roughly equivalent to 60 kbar pressure were obtained. Modelling changes that occur i n ' -quartz at t h i s pressure, d(Si-Ob) was kept constant while decreasing the SiOSi angle from 144° tb 134° (path A-C); path B-C shows the x associated' with an increment of 60 kbar pressure. -secASiOSi V I . CONCLUSIONS M o l e c u l a r o r b i t a l t h e o r y i s a bonding f o r m a l i s m based upon quantum m e c h a n i c a l p r i n c i p l e s and has been a p p l i e d t o m i n e r a l o g i c a l s t u d i e s of e q u i l i b r i u m m o l e c u l a r geometry, e l e c t r o n i c charge d i s t r i b u t i o n s , e l e c t r o n i c s p e c t r a and f o r c e c o n s t a n t c a l c u l a t i o n s . To d a t e , t h e s e s t u d i e s have been l i m i t e d t o one atmosphere p r e s s u r e . W i t h the ever i n c r e a s i n g i n t e r e s t i n u l t r a - h i g h p r e s s u r e phases and mantle m i n e r a l o g y , bonding s t u d i e s of m o l e c u l a r groups a t s i m u l a t e d h i g h p r e s s u r e can be an i n v a l u a b l e a i d t o u n d e r s t a n d i n g h i g h p r e s s u r e c r y s t a l c h e m i s t r y , bond e n e r g e t i c s and e l e c t r o n i c s p e c t r a . In a d d i t i o n , such s t u d i e s w i l l e nable us t o s i m u l a t e p r e s s u r e s beyond the l i m i t s of c u r r e n t e x p e r i m e n t a l t e c h n o l o g y . T h i s i n v e s t i g a t i o n i s devoted t o the study of e q u i l i b r i u m S i - 0 bond l e n g t h s , S i O S i a n g l e s and S i - 0 f o r c e c o n s t a n t s w i t h i n c r e a s i n g p r e s s u r e . A l t h o u g h the method of a p p l y i n g p r e s s u r e i s r a t h e r crude i n t h a t h e l i u m atoms a r e used t o a p p l y a d i r e c t e d s t r e s s a x i a l w i t h the S i - 0 b r i d g i n g bond l e n g t h , we f e e l the r e s u l t s a re r e a s o n a b l e a p p r o x i m a t i o n s of expected t r e n d s . For example, w i t h i n c r e a s i n g p r e s s u r e the S i - 0 bond l e n g t h and S i O S i a n g l e d e c r e a s e 0.3% and 4.5% , r e s p e c t i v e l y , up t o 60 kbar p r e s s u r e which compares w e l l w i t h the 0.3% and 6.6% d e c r e a s e observed i n c - q u a r t z ( L e v i e n e_t a_l. , 1980). F u r t h e r m o r e , the l i n e a r c o r r e l a t i o n of S i - 0 bond l e n g t h and - s e c ( S i O S i ) , known t o oc c u r a t one atmosphere, h o l d s a t i n c r e a s e d p r e s s u r e ; t h i s t r e n d i s a l s o observed i n c o e s i t e a t h i g h p r e s s u r e s . Symmetric S i - 0 s t r e t c h i n g and S i O S i bending f o r c e c o n s t a n t s show a p e r c e n t a g e i n c r e a s e i n the r a t i o of 1:6 up t o an e s t i m a t e d p r e s s u r e of 140 kbars which i s i n keeping w i t h the r e l a t i v e d e c r e a s e i n d ( S i - O b ) and the S i O S i a n g l e . E x p e r i m e n t a l l y d e t e r m i n e d s t r e t c h i n g and bending f o r c e c o n s t a n t s i n s i l i c a t e s a t h i g h p r e s s u r e are s p a r s e . F e r r a r o e t a l . (1972) and F e r r a r o and Manghnani (1972) have i n v e s t i g a t e d the i n f r a r e d s p e c t r a of c - q u a r t z , f u s e d s i l i c a , P y r e x , Vycor and a v a r i e t y of sodium s i l i c a t e g l a s s e s a t p r e s s u r e s up t o 58.8 kbar. The a b s o r p t i o n bands a t t r i b u t e d t o S i - O - S i s t r e t c h v i b r a t i o n s show, i n g e n e r a l , a p o s i t i v e dependence w i t h p r e s s u r e i n d i c a t i n g a c o r r e s p o n d i n g i n c r e a s e i n the s t r e t c h i n g f o r c e c o n s t a n t . S i m i l a r l y the mixed bending fr e q u e n c y of the S i O S i and OSiO a n g l e s shows a p o s i t i v e dependence w i t h p r e s s u r e f o r a - q u a r t z and the sodium s i l i c a t e g l a s s e s ; the p o s i t i v e p r e s s u r e dependence noted f o r t h i s f r e q u e n c y p r i m a r i l y r e f l e c t s the change i n the S i O S i a n g l e ( F e r r a r o e t a l . , 1972) and i n d i c a t e s t h a t the S i O S i bending f o r c e c o n s t a n t i s i n c r e a s i n g w i t h p r e s s u r e . A l t h o u g h t h i s study has f o c u s e d on the H 6 S i 2 0 7 c l u s t e r , i t r e p r e s e n t s the i n i t i a l i n s t a l l m e n t i n a s e r i e s of s t u d i e s on the c o m p r e s s i b i l i t i e s of g e o l o g i c a l l y i m p o r t a n t m e t a l - oxygen p o l y h e d r a . Work i s c u r r e n t l y i n p r o g r e s s on the H 4SiO„ and Hg-AlO/," 1 t e t r a h e d r a and we a r e c a l c u l a t i n g f o r c e c o n s t a n t s , p o l y h e d r a l b u l k m o d u l i , Kp, as w e l l as the f i r s t d e r i v a t i v e of Kp w i t h r e s p e c t t o p r e s s u r e , d(Kp)/dP. F u t u r e work w i l l be devoted t o f o r c e c o n s t a n t , Kp and d(Kp)/dP d e t e r m i n a t i o n s f o r oxyanion c l u s t e r s of magnesium, aluminum and s i l i c o n i n o c t a h e d r a l c o o r d i n a t i o n . U l t i m a t e l y we hope t o approximate the b u l k modulus of a s o l i d phase a t h i g h p r e s s u r e t h r o u g h computed Kp and bending f o r c e c o n s t a n t s . REFERENCES B i n k l e y , J.S., R. W h i t e s i d e , P.C. H a r i b a r a n , R. Seeger, W.J. Hehre, W.A. L a t h a n , M.D. Newton, R. D i t c h f i e l d , and J.A. P o p l e , 1978, G a u s s i a n 76- an ab i n i t i o m o l e c u l a r o r b i t a l program: Quantum Program Chemical Exchange, Bl o o m i n g t o n , IN. Brown, G.E., G.V. Gibbs and P;.H. R i b b e , 1969, The n a t u r e and v a r i a t i o n i n l e n g t h of the S i - 0 bond and A l - 0 bonds i n framework s i l i c a t e s : Am. M i n e r a l . , 54 , 1044-1061. Cameron, M., S. Sueno, C T . P r e w i t t and J . J . P a p i k e , 1973, High temperature c r y s t a l c h e m i s t r y of a c m i t e , d i o p s i d e , h e d e n b e r g i t e , j a d e i t e , spodumene and u r e y i t e : Am. M i n e r a l . , . 58 , 594-618.' C o l l i n s , J.B., P. Von R. S c h l e y e r , J.S. B i n k l e y and J.A. P o p l e , 1976, S e l f - c o n s i s t e n t m o l e c u l a r o r b i t a l methods. X V I I . Geometries and b i n d i n g e n e r g i e s of second row m o l e c u l e s . A comparison of t h r e e b a s i s s e t s : J o u r . Chem. Phys., 64 , 5142-5151. De Jong, B.H.W.S. and G.E. Brown, 1980, The p o l y m e r i z a t i o n of s i l i c a t e and a l u m i n a t e t e t r a h e d r a i n g l a s s e s , m e l t s and aqueous s o l u t i o n s - I . E l e c t r o n i c s t r u c t u r e of H 6 S i 2 0 7 , H 6 A l S i 0 7 1 " and H 6 A 1 2 0 7 2 "" : Geochim. Cosmoehim. A c t a , 4_4 , 491-511 . Farmer, V.C., ed., 1974, The i n f r a - r e d s p e c t r a of m i n e r a l s : M i n e r a l . Soc. London, 539 p. F e r r a r o , J.R. and M.H. Manghnani, 1972, I n f r a r e d a b s o r p t i o n s p e c t r a of sodium s i l i c a t e g l a s s e s a t h i g h p r e s s u r e s : J o u r . A p p l . Phys., 43 , 4595-4598. F e r r a r o , F.R., M.H. Manghnani and A. Q u a t t r o c h i , 1972, I n f r a r e d s p e c t r a of s e v e r a l g l a s s e s a t h i g h p r e s s u r e s : Phys. Chem. G l a s s e s , J_3, 116-121. G i b b s , G.V., E.P. Meagher, M.D. Newton and D.K. Swanson, i n p r e s s , A comparison of e x p e r i m e n t a l and t h e o r e t i c a l bond l e n g t h and an g l e v a r i a t i o n s f o r m i n e r a l s , i n o r g a n i c s o l i d s and m o l e c u l e s : 34 p. I_n O'Keeffe, M. and A. N a v r o t s k y ( e d s . ) , S t r u c t u r e and bonding i n c r y s t a l s : Academic P r e s s , New York. G i b b s , G.V.,C.T. P r e w i t t and K.J. B a l d w i n , 1977, A stu d y of of s t r u c t u r a l c h e m i s t r y of c o e s i t e : Z e i t . K r i s t a l l . , 145 , 108-123. G i l l e s p i e , R.J. and E.A. Robinson, 1964, C h a r a c t e r i s t i c 50 f r e q u e n c i e s of compounds c o n t a i n i n g S i - O - S i , P-O-P, S-O-S and v i b r a t i o n a l C l - O - C l b r i d g i n g groups: Can. J o u r . Chem., 42 ,2496-2503. Hazen, R.M. and L.W. F i n g e r , 1979, Bulk modulus-volume r e l a t i o n s h i p f o r c a t i o n - a n i o n p o l y h e d r a : J o u r . Geophys. Res., 84 , 6723-6728. Hehre, W.J., R.F. St e w a r t and J.A. P o p l e , 1969, S e l f - c o n s i s t e n t m o l e c u l a r o r b i t a l methods. I . Use of Gaussian e x p a n s i o n s of S l a t e r - t y p e atomic o r b i t a l s : J o u r . Chem. Phys. , 5J_ , 2657-2664. H e r z b e r g , G., 1945, I n f r a r e d and raman s p e c t r a of p o l y a t o m i c m o l e c u l e s , v o l . 2: D. Van N o s t r a n d Co., New York, 632 p. J e z o w s k a - T r z e b i a t o w s k a , B., J . Hanuza and W. W o j c i e c h o w s k i , 1967, I n f r a - r e d and v i b r a t i o n a l f r e q u e n c i e s of the X-O-X bonds f o r the I V t h p e r i o d i c group of elements: S p e c t r o c h i m . A c t a , 23A , 2631-2636. J o r g e n s e n , J.D., 1978, Compression mechanisms i n a-quartz s t r u c t u r e s - S i 0 2 and Ge0 2: J o u r . A p p l . Phys., 49 , 5473- 5478. L a z a r e v , A.N., 1972, V i b r a t i o n a l s p e c t r a and s t r u c t u r e s of s i l i c a t e s , t r a n s l a t e d from R u s s i a n : C o n s u l t a n t s Bureau, New York, 302 p. L e v i e n , L. and C.T. P r e w i t t , 1981, H i g h - p r e s s u r e c r y s t a l s t r u c t u r e and c o m p r e s s i b i l i t y of c o e s i t e : Am. M i n e r a l . , 66 , 324-333. L e v i e n , L., C.T. P r e w i t t and D.J. Weidener, 1980, S t r u c t u r e and e l a s t i c p r o p e r t i e s of q u a r t z a t p r e s s u r e : Am. M i n e r a l . , 65 , 920-930. L o u i s n a t h a n , S.J. and G.V. G i b b s , 1972, V a r i a t i o n of S i - 0 d i s t a n c e s i n o l i v i n e s , s o d a m e l i l i t e and sodium m e t a s i l i c a . t e as p r e d i c t e d by s e m i - e m p i r i c a l m o l e c u l a r o r b i t a l c a l c u l a t i o n s : Am. M i n e r a l . , 57 , 1643-1663. Lyon, R.J.P., 1962, M i n e r a l s i n the i n f r a r e d : S t a n f o r d R e search I n s t i t u t e , C a l i f o r n i a . McWeeney, R., 1979, Cou l s o n ' s v a l e n c e : O x f o r d U n i v . P r e s s , O x f o r d , 434 p. Meagher, E.P., J.A. T o s s e l l and G.V. G i b b s , 1979, A CNDO/2 m o l e c u l a r o r b i t a l study of the s i l i c a polymorphs q u a r t z , c r i s t o b a l i t e and c o e s i t e : Phys. Chem. M i n . , 4 , 11-21. M i l k e y , R.G., 1960, I n f r a r e d s p e c t r a of some t e c t o s i l i c a t e s : Am. M i n e r a l . , 45 , 990-1003. Moenke, H., 1962, M i n e r a l s p e k t r e n , I . : A kademie-Verlag, B e r l i n . Moenke, H., 1966, M i n e r a l s p e k t r e n , 1 1 . : A kademie-Verlag, B e r l i n . M u l l i k e n , R.S., 1955, E l e c t r o n i c p o p u l a t i o n a n a l y s i s on LCAO- MO m o l e c u l a r wave f u n c t i o n s . I . : J o u r . Chem. Phys., 23 , 1833-1840. Newton, M.D. and G.V. G i b b s , 1980, Ab i n i t i o c a l c u l a t e d g e o m e t r i e s and charge d i s t r i b u t i o n s f o r H^SiO, and H 6 S i 2 0 7 compared w i t h e x p e r i m e n t a l v a l u e s f o r s i l i c a t e s and s i l o x a n e s : Phys. Chem. M i n . , 6 , 221-246. Newton, M.D., W.A. L a t h a n , W.J. Hehre and J.A. P o p l e , 1970, S e l f - c o n s i s t e n t m o l e c u l a r o r b i t a l methods. V. Ab i n i t i o c a l c u l a t i o n of e q u i l i b r i u m g e o m e t r i e s and q u a d r a t i c f o r c e c o n s t a n t s : J o u r . Chem. Phys., 52 , 4064-4072. Newton, M.D., M. O'Keeffe and G.V. G i b b s , 1980, Ab i n i t i o c a l c u l a t i o n of i n t e r a t o m i c f o r c e c o n s t a n t s i n H 6 S i 2 0 7 and the b u l k modulus of c - q u a r t z and c - c r i s t o b a l i t e : Phys. Chem. M i n . , 6 , 305-312. O h a s h i , Y. and C.W. Burnham, 1972, E l e c t r o s t a t i c and r e p u l s i v e e n e r g i e s of the M1 and M2 c a t i o n s i t e s i n pyrox e n e s . J o u r . Geophys. Res., 7_7 , 5761-5766. P a p i k e , J . J . , ed., 1969, Pyroxenes.and a m p h i b o l e s . C r y s t a l c h e m i s t r y and phase d e t e r m i n a t i o n s : M i n e r a l . Soc. Am. S p e c i a l Paper, 2. P a u l i n g , L., 1980, The na t u r e of the s i l i c o n - o x y g e n bonds: Am. M i n e r a l . , 65 , 321-323. P o p l e , J.A., D.P. S a n t r y and G.A. S e g a l , 1965, Approximate s e l f - c o n s i s t e n t m o l e c u l a r o r b i t a l t h e o r y . I . I n v a r i a n t p r o c e d u r e s : J o u r . Chem. Phys., 43̂  , S129-S135. Ross, S.D., 1972, I n o r g a n i c i n f r a r e d and raman s p e c t r a : McGraw- H i l l Book Co. (UK) Lmtd., London, 448 p. T o s s e l l , J.A., 1973, M o l e c u l a r o r b i t a l i n t e r p r e t a t i o n of x-ray emmission and ESCA s p e c t r a l s h i f t s i n s i l i c a t e s : J o u r . Phys. Chem. S o l i d s , 34 , 307-319. T o s s e l l , J.A., 1979, D i v e r s e c h e m i c a l bond t y p e s i n m i n e r a l s : T r a n s . Am. C r y s t a l l . A s s o c . , j_5 , 47-63. T o s s e l l , J.A. and G.V. G i b b s , 1977, M o l e c u l a r o r b i t a l s t u d i e s of g e o m e t r i e s and s p e c t r a of m i n e r a l s and i n o r g a n i c compounds: Phys. Chem. M i n . , 2 , 21-57. 52 W h i t t a k e r , E.J.W., 1971, Madelung e n e r g i e s and s i t e p r e f e r e n c e s i n a m p h i b o l e s , 1 : Am. M i n e r a l . , 56 , 980-996.

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