Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Ab initio SCF MO study of H₆SI₂O₇ at simulated high pressure Ross, Nancy Lee 1981

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1981_A6_7 R68.pdf [ 2.59MB ]
Metadata
JSON: 831-1.0064299.json
JSON-LD: 831-1.0064299-ld.json
RDF/XML (Pretty): 831-1.0064299-rdf.xml
RDF/JSON: 831-1.0064299-rdf.json
Turtle: 831-1.0064299-turtle.txt
N-Triples: 831-1.0064299-rdf-ntriples.txt
Original Record: 831-1.0064299-source.json
Full Text
831-1.0064299-fulltext.txt
Citation
831-1.0064299.ris

Full Text

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., 43^  , 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. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0064299/manifest

Comment

Related Items