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An M.O. study of the stereochemistry and energy of three-repeat single silicate chains Roussy, Marie Monique 1982

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M.O. STUDY OF THE STEREOCHEMISTRY AND ENERGY OF THREE-REPEAT SINGLE SILICATE CHAINS by MARIE MONIQUE ROUSSY B.Sc,  M c G i l l U n i v e r s i t y , 1978  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in  THE  FACULTY 'OF GRADUATE STUDIES  DEPARTMENT OF GEOLOGICAL SCIENCES  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1982  Marie Monique Roussy, 1982  >  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 of  requirements f o r an advanced degree a t the  the  University  of 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 it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  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 o f 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  department or by h i s or her  the head of  representatives.  my  It is  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department of  Geological Sciences  The U n i v e r s i t y of B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6  (3/81)  September 3 r d . 1982  Columbia  written  ii  ABSTRACT  Hypothetical regular  SiO«  three—repeat  single  t e t r a h e d r a were generated  to study the p o s s i b l e conformations energy  chains  and  consisting  with a computer program their  total  potential  by the CNDO/2 molecular o r b i t a l method. C a l c u l a t i o n s were  restricted  to  those chains with / S i - O - S i between 110° and  with an average addition,  of 140° and /(O-O-O) of  tetrahedra  r e s t r i c t e d to e i t h e r  single  or  double—eclipsed  conformation.  chains  with  l e a s t one are  successive  between 90° and  pairs  These  of  Those  staggered p a i r of not,  as  the  tetrahedra  y e t , observed  180°.  180° In  i n each chain were or  staggered  l a r g e s t d ( S i — S i ) and at have  a  lower  energy.  i n nature presumably because  t h e i r packing does not accomodate the i n t e r s t i a l c a t i o n s .  iii  TABLE OF CONTENTS  ABSTRACT LIST OF TABLES  i i iv  LIST OF FIGURES  v  ACKNOWLEDGEMENT  vi  I.  INTRODUCTION  1  11 .  THE CNDO/2 METHOD  2  1 . THEORY  2  2. CNDO/2 APPROXIMATION  3  3. CNDO/2 PARAMETRIZATION  4  I I I . MODELS OF THREE-REPEAT CHAINS  6  IV.  V.  1. DESCRIPTION OF CHAIN CONSTRUCTION  6  2. DESCRIPTION OF FIXED PAIRS  6  CLASSIFICATION OF CHAINS  8  1. DEFINITION OF SYMBOLS  8  2. REPRESENTATIVE CHAINS  11  DISCUSSION OF ENERGY CALCULATIONS  '13  1. SINGLE CHAIN SELECTION  13  2. RESULTS OF ENERGY CALCULATIONS  18  3. PACKING OF CHAINS  23  CONCLUSION  27  V I I . REFERENCES  29  VI.  iv  LIST OF TABLES  Table  Page  Unique s e t t i n g s f o r the r o t a t i o n angles < > j , 9 , <fr and 6 that y i e l d double or s i n g l e — e c l i p s e d and staggered p a i r s .  10  Summary of symbols used i n d e s c r i b i n g three—repeat single c h a i n s . These represent a l l the p o s s i b l e comformations of the double or s i n g l e — e c l i p s e d and staggered p a i r s of t e t r a h e d r a i n a c h a i n .  12  III. List of three—repeat chains composed of two f i x e d p a i r s of t e t r a h e d r a . The p a i r quoted i n parantheses represents the c l o s e s t approximation to the standard o r i e n t a t i o n of the t h i r d and f o u r t h t e t r a h e d r a . The / S i - O - S i vary between 120° and 180° with an average of 140°. The / ( 0 - 0 - 0 ) vary between 90° and 180°.  14  I.  2  2  3  3  II.  br  IV.  /Si-O-Si and /(O-O-O) w o l l a s t o n i t e (W) . ** b  f o r bustamite  (B) and  17  V  LIST OF FIGURES  Figure  Page  Original configuration showing 2/m point group a f t e r r o t a t i o n by angles 30°, f = 40°, (j> = 10° 3  3  of the three—repeat chain symmetry and the same c l u s t e r C = 40°, 4> = 35°, 0 = and 0 = 45°. 2  2  2  3  7  Double—eclipsed, single—eclipsed and staggered p a i r s of t e t r a h e d r a with energies -137.9974, -137.9977 and -137.9983 A.U. r e s p e c t i v e l y .  9  Three segments of the same chain whose e n e r g i e s are -256.172, -256.170 and -256.170 A.U. r e s p e c t i v e l y .  19  T o t a l energy (A.U.) against average d ( S i — S i ) (A), those chains with the l a r g e s t d ( S i - S i ) have the lowest energy.  21  Total energy (A.U.) against each group of c h a i n s .  (A) f o r '  22  Wollastonite (Buerger and Prewitt,1961) and i t s c l o s e s t approximation with t o t a l energy -256.158 A.U.  24  Chains containing a combination of double or s i n g l e — e c l i p s e d p a i r s with energies -256.164, -256.163 and -256.163 A.U. r e s p e c t i v e l y .  25  Chains c o n t a i n i n g a staggered p a i r of t e t r a h e d r a with either a double or s i n g l e eclipsed or another staggered p a i r . Energies are -256.165, -256.167 and -256.164 A.U. r e s p e c t i v e l y .  26  average d ( S i — S i )  vi  ACKNOWLEDGEMENT  I am g r a t e f u l to support  throughout  Dr E.P  this  Meagher  f o r h i s guidance  and  study, T h i s work was supported by NSERC  grant 67-7061. Table II was compiled  from data  calculated  by  Ms.  Irene  King. I  also  illustrations  thank  Mr.  i n the t e x t .  Gord  Hodge  f o r draughting  a l l  the  1  I. INTRODUCTION  Only one type of found  in  nature  three—repeat  in  single  silicate  chain  the minerals w o l l a s t o n i t e , bustamite, the  p e c t o l i t e — s e r a n d i t e s e r i e s , s c h i z o l i t e and s o r e n s o n i t e . study  of  the  tetrahedral  role  chains  sorensonite,  of  octahedral  in  Ohashi  each  and  of  cations  the  Finger  above  (1978)  chemical The  of  three—repeat  single  minerals,  occur as a  which  are  chains by Meagher  function  between  125° and 150° and b r i d g i n g  between  125° and 180° i s lower  E a r l i e r work on  those chains with b r i d g i n g Si—O—Si angles, / S i — O — S i ,  conformations.  Meagher  found  oxygen  apparently l e s s important This  angles,  /(O—O—O)^ ,  than that of chains with any oter that the chains of lowest  are those with both / S i - O - S i approaching  energy  135°. The / ( 0 - 0 - 0 )  in determining  the  energy  of  br  is  these  study was t h e r e f o r e l i m i t e d to those three—repeat  s i n g l e s i l i c a t e chains with / S i - O - S i between average  types  (1980) has shown that the p o t e n t i a l  of  of  140°  and /(0-0-0)  reducing the number of  chains  reduce  of  the  number  chains  tetrahedra  that  were  either  with  between 90° and 180°, thus  fcr  to  110° and 180°  be  considered. and  to  To  facilitate  i d e n t i f i c a t i o n each was looked upon as composed of of  except  stereochemically  t h e i r r e l a t i v e energy.  energy  an  i n the packing of  i s to d e f i n e the d i f f e r e n t  chains  p o s s i b l e and to determine  chains.  a  variations.  purpose of t h i s study  two—repeat  From  found that only small  changes i n the comformation of the chains of  is  three  further their pairs  s i n g l e or d o u b l e — e c l i p s e d or  2  staggered.  I I . THE CNDO/2 METHOD  1. Theory  The t o t a l energy Schrodinger  E of an atomic  H  i s given  by the  equation Hlf  where  cluster  = E l\>  (1 )  i s the Hamiltonian Operator  function. This  equation  cannot  be  and  solved  i s the t o t a l wave for  many—electron  problems. The 1930)  Hartree-Fock  Operator,  i s an appproximation F = H  where H  i s called  represents  t  the one—center  the a t t r a c t i o n  between  and  describe integral  electrons K^j  1928 and Fock,  given by  + J^' - 1/2K^  o r b i t a l and the n u c l e i  coulomb  (Hartree,  of the Hamiltonian  molecular K^j  F,  or  core  Hamiltonian  an  electron  the  terms J 'j t  interactions. repulsion  i n the i — t h and j — t h molecular o r b i t a l s  and  i n the i — t h  i n the c l u s t e r . The  the two—electron representing  (2)  J  V J  '  between  i s the two  respectively.  i s the exchange i n t e g r a l r e p r e s e n t i n g the decrease  i n energy  a s s o c i a t e d with two e l e c t r o n s having the same s p i n . The  t o t a l wave f u n c t i o n i s an  molecular combination  orbitals  antisymmetrized  product  of  which Roothan (1951) r e p r e s e n t s by a l i n e a r  of n atomic  orbitals,  S  , (LCAO) (3)  3  For orthonormal  atomic o r b i t a l s ^  , <|>^ the o v e r l a p matrix  i s given by s  n-  =  f  $r  ( s )  ^  (  s  )  a  r  (  s  )  ( 4 )  where d T i s the volume element a s s o c i a t e d with e l e c t r o n s. The  best approximation  of the energy of an atomic  cluster,  i , i s given by the V a r i a t i o n Theorem. J l//*Hy/dT  <L  E =  (5)  the c o e f f i c i e n t s c^p that correspond to the minimun energy s a t i s f y the set of simultaneous  where (  5  must  equations  M>  0  C  (6)  i s a s o l u t i o n t o the s e c u l a r equation | F  - ES ^| = 0  f f  (7)  p  2. CNDO/2 Approximation  The  Complete Neglect of D i f f e r e n t i a l Overlap  (CNDO) method  was introduced by Pople, Santry and Segal i n 1965. associated  with  the  electron  repulsion  The  between  o r b i t a l s with d i f f e r e n t p r i n c i p a l quantum number i s be n e g l i gaajbj ilce,,  overlap molecular  assumed  to  i.e.,  *'(s) Ifj (s) r ;  1 T  Vj/*(t) Vj/(t) dT  5  dT  T  = 0 (8)  for  i # j and k  ±  1  In a d d i t i o n the corresponding o v e r l a p i n t e g r a l s are neglected i n the  normalization  of  the  molecular  orbitals.  The remaining  4  electron—repulsion  i n t e g r a l s of the form  .V 4>,*>  (s) r  *-  ( t )dr  d7:  =  (9) are e v a l u a t e d . between  This  term  represent  the  valence e l e c t r o n s assumed to belong  average  interaction  to a s—type  orbital  on atoms a and b r e s p e c t i v e l y 3. CNDO/2 P a r a m e t r i z a t i o n  Each d i a g o n a l matrix element H one—center  and a two—center  cn  be  separated  into  a  c o n t r i b u t i o n s of the form  = U£" - £  V^  (j^on a  b  (10)  where  un =  -1/2  (  A; )  ij+  -  (  z-*-)  r(11)  and  1/2  ( Ii. + A  )  is  electronegativity,where I  the  Mulliken  definition  of  i s the i o n i z a t i o n p o t e n t i a l and A  the e l e c t r o n i c a f f i n i t y a s s o c i a t e d with  <^  .  Z^  is  the  is core  charge of atom a. U^i represents the t o t a l energy of an e l e c t r o n belonging  to the atomic  orbital  <^  centered on atom a. The term  V„,J> given by «cfc = fe *o.l  v  (12)  z  which r e p r e s e n t s the i n t e r a c t i o n of any valence e l e c t r o n on atom a with the core of atom b. For  two  atomic  orbital  t  on  atoms  r e s p e c t i v e l y the o f f - d i a g o n a l matrix element H^j becomes  a  and b  5  Htj  where  /-^t j  parameter  is  Ay = / U  =  called  the  and  K  /^y  small  =  integral.  +  1/2K(  The  bonding  )  (14)  0 . 7 5 i f e i t h e r a or b i s a second  are found by b e s t — f i t t i n g  to f u l l SCF  row  element  ./^L  calculations  for  molecules. To  summerize, d i a g o n a l elements  take the *u  resonnance  / ^ l i s given by  Ai= where  (is)  s+k  =  u  of the Hartree—Fock  matrix  form  t-:  <p„*  -  p-)  1/2  K~  X,  +  [-Q^A  +  (z  -  b  >] (15)  The net charge,  , i s given by Q  = f> - t,b p  z  b  (  where the gross e l e c t r o n i c p o p u l a t i o n on atom b, Pj, , b  is  1  6  )  given  by  and the d e n s i t y matrix, p|,r i ^ ^ g i v e n by p  p  The  q u a n t i t y -Q  to  the  b  X^t  z  due  total -  f> =  2  S  (18)  represents the e f f e c t of the p o t e n t i a l  charge  on  represents  atom the  b.  The  penetration  integral,  d i f f e r e n c e between the p o t e n t i a l s  to the valence e l e c t r o n s on the core of the n e u t r a l atom  The o f f - d i a g o n a l elements F  All  cj  (Dobash, 1 9 7 4 ) with (1965)  for  take the  - A ^ S p ^ -  calculations  b.  form 1  / f>^  (19)  2p  were c a r r i e d out with the program CNINDO  the  the f i r s t  due  parametrization  row elements  of  Pople  and  and that of Santry and  Segal Segal  6  (1967) f o r the second row elements.  Ill.  MODELS OF THREE-REPEAT CHAINS  1. D e s c r i p t i o n of chain c o n s t r u c t i o n  A  computer  four—membered, of  program  was  equal shows  d(Si—0).  In  as a f u n c t i o n  180° and the c l u s t e r has the p o i n t an  atomic  cluster  before  A  2  2  brings  rotation  brings  into them  a  angle  <^>  2  and ©  t h i r d tetrahedron  2  the  Si—0  into  a  2/m.  rotation  t r a n s l a t i n g the f i r s t non-bridging  angle  oxygen atoms on the f i r s t and pair,  single—eclipsed  a  clockwise  pair. Either a  r o t a t i o n about the -Si—0  bonds  r e s p e c t i v e l y c r e a t e a staggered p a i r . The  i s a l s o r o t a t e d by an angle £  i t s own S i - 0  Fig 1  including  r o t a t i o n by an  3  about  which i s the p r e v i o u s l y r o t a t e d y — a x i s and by angles ^ about  bond  i s free to r o t a t e about the  double—eelipsed  clockwise or a counterclockwise by  any  tetrahedron remains f i x e d  counterclockwise  the four non—bridging  second t e t r a h e d r a  of  symmetry  and a f t e r  a l l times. The second tetrahedron  y — a x i s by an angle (f . £  calculates  the o r i g i n a l c o n f i g u r a t i o n , both / S i — O - S i  r e f e r e n c e and r o t a t i o n axes. The f i r s t at  that  three—repeat c h a i n s t a r t i n g with the c o o r d i n a t e s  three t e t r a h e d r a expressed  length,  written  bonds. The f o u r t h tetrahedron  y'—axis 3  and ©  i s added by  tetrahedron along a vector connecting  oxygen atoms on the. f i r s t  2. D e s c r i p t i o n of the f i x e d  pairs  3  and t h i r d t e t r a h e d r a .  the  8  In  order  to  simplify  the i d e n t i f i c a t i o n of the types of  three—repeat chains each i s looked upon pairs  of  tetrahera.  Fig  2  move  atoms  on  closer  each  to  single—eclipsed  if  other  only  moves s i m i l a r l y . The p a i r rotated one  so  is  narrowing  oxygen  i s staggered  other  Table  if  one  tetrahedron  and  angles ^>  2  ,  O  the 2>  s i n g l e — e c l i s e d and in  a  these  first  &  3  and  0  that  3  staggered p a i r s . The  yields  and fixed  constant.  c h o i c e of unique s e t t i n g s f o r the  <^> 3  three  / S i - O — S i . No  the  2  is  bisecting  of t e t r a h e d r a i n a chain since these can be  shows  or  tetrahedron  the r o t a t i o n angles ? , 4> 21 <J> 3 and  I  two  atom on each tetrahedron  v a r i a t i o n of the above types are c o n s i d e r e d f o r  by keeping  if  /Si—O-Si  that one oxygen atom r e s i d e s on the plane  pairs  three  conformation  atoms move toward each other with narrowing  second  of  double—eclipsed  i n an e c l i p s e d  with  one  of the /0—Si—0 on the  oxygen  It  tetrahedron  each  composed  shows three standard o r i e n t a t i o n s  between the members of a p a i r . oxygen  as  rotation  double  or  t h i r d p a i r of t e t r a h e d r a  chain depends on the t r a n s l a t i o n of the f i r s t  tetrahedron  so intermediate types are p o s s i b l e .  IV. CLASSIFICATION OF CHAINS  1. D e f i n i t i o n of symbols  Each of the p a i r s of t e t r a h e d r a d e s c r i b e d i n chapter  the  previous  i s i d e n t i f i e d by a l e t t e r ; D for d o u b l e - e c l i p s e d , S f o r  single-eclipsed  and  St  for  staggered.  Furthermore  the  DOUBLE ECLIPSED PAIR  respectively  SINGLE ECLIPSED PAIR  STAGGERED PAIR  10  Table I. Unique s e t t i n g s f o r the r o t a t i o n angles 1 and O that yield double or s i n g l e - e c l i p s e d and pa i r s . 2  3  F i r s t pair  ^ 2  0  2, 3 staggered  Second p a i r  0  2  ^3  Double—eclipsed  ©3  pairs  0°  0°  0°  0°  0°  120°  120°  0°  Single—eelipsed  pairs  60°  60°  60°  60°  60°  180°  180°  60°  Staggered  pairs  0°  60°  60°  0°  60°  0°  0°  60°  0°  180°  180°  0°  11  relationship will  of  the b r i d g i n g oxygen atoms from which the chain  be extended, i . e . , a l l oxygen atoms other than 0, on F i g u r e  2, with respect to the S i — 0 — S i plane of the p a i r a number. On F i g u r e 2 0  is  b n  always  oxygen  i s d e s c r i b e d by  atom  0,.  These  symbols are summerized i n Table I I . These  symbols  f o l l o w s . The both  one  symbols D,,  bridging  p a i r . The  can  be  grouped  S, and St,  into  represent  symbols D ,  S,  2  pair.  the  case  where  oxygen atoms are i n the S i — 0 — S i plane of a given St  2  2  and S t  r e f e r to  5  of the b r i d g i n g oxygen atom i s not  given  four c a t e g o r i e s , as  The  last  the  case  where  i n the S i — 0 — S i plane of a  two c a t e g o r i e s i n c l u d e the p a i r s i n which  both b r i d g i n g oxygen atoms are o f f the S i — 0 — S i plane of a pair.  The  symbols  D, 3  S  and S t  3  designate the case where the  3  b r i d g i n g oxygen atoms are both on the same s i d e of plane  while  the symbols D , a  given  S„ and  St„  the b r i d g i n g oxygen atoms are on e i t h e r  the  Si—O—Si  r e f e r to the case where side of t h i s  plane.  2. R e p r e s e n t a t i v e chains  The work of Meagher (1980)  and  i n d i c a t e s that nearest neighbour in determining Accordingly,  Newton  and  i n t e r a c t i o n s are most  defined  to  which by  restrictions  (1980)  important  the comformation of b r i d g i n g s i l i c a t e t e t r a h e d r a . decrease  the number of chains to be c o n s i d e r e d  only those c h a i n s with adjacent t e t r a h e d r a having /(0—0—0)^  Gibbs  fall  into  the  these  authors  will  were  imposed  on  region be  the  /Si—0—Si  of minimum energy  computed. chains.  The  The  and as  following  / S i — O — S i are  12  Table I I . Summary of symbols used in d e s c r i b i n g three-repeat s i n g l e c h a i n s . These represent a l l the p o s s i b l e comformations of the double or s i n g l e - e c l i p s e d and staggered p a i r s of t e t r a h e d r a in a c h a i n .  Symbols  Choice of 0 it r — br Double—eclipsed  D, D D D D D D,  pairs 0 0 0 0„ 0 0 0«  and and and and and and and  0 either 0 either 0 0 0 0 0  0 and 0 and 0 and 0 and On and 0 and 0« and  0 either 0 either 0 0 0 0 0  0„ 0 0 0 00 0«  either 0 0 0 0 0 either 0  2  (a) (b) (a) (b) (a) (b)  2  2  3  3  4  Single—eclipsed S, S 5 5 S S S„ 2  3  3  a  Staggered St, St St St St« St„ St 2  3  3  5  2  3 3  7  or or  0„ 0  6  or or  0„ 0  2  or  0  3  6  or  0  7  3 6  7  5  6  5  6  pairs (a) (b) (a) (b) (a) (b)  2  7  2  5  2  3  3  5  3  7  6  7  7  6  pairs  (a) (b) (a) (b)  5  2  3  2  3  and and and and and and and  O5  7  6 6  7  13  between 110° and  180° with an average of  90°  and  between  i.e.,  the d(0—0) i n a r e g u l a r tetrahedron where the  A  list  of  of i o n i c  three—repeat  The  The  vary  are equal to the sum  180°.  140°.  /(0-0-0).  s h o r t e s t d ( 0 - 0 ) ^ i , i s 2.63  A  r  radii  Si—0  bonds  (Shannon and P r e w i t t , 1969).  chains composed of two  f i x e d p a i r s as  d e s c r i b e d above i s given on Table I I I . Table IV shows wollastonite  a  list  (Buerger  of  and  2  3  Yamanska, 1976) and OH"  Finger,  and  series  (Prewitt,  9  and  are not  group i n the c h a i n . The  and  taking  (Metcalf—Johansen these compounds  1967;  Takeuchi,  presence  /(0-0-0)  .  b  The  average between 142° and and  1976)  very  /Si-O-Si  Kudoh  pectolite,  9  and  (Ohashi  of a water molecule  sorensonite,  a  3  i n c l u d e d because of the presence  and H a z e l l , have  bustamite 2  a hydrogen bridge with oxygen atoms bonded to the precludes  for  br  pectolite(Ca NaHSi 0 )-  s c h i z o l i t e , a manganoan  1978)  /(0-0-0) ,  P r e w i t t , 1961)  (Peacor and Buerger, 1962). The serandite(Mn NaHSi 0 )  /Si-O-Si  are  silicon 2  range  between  145°. The  forming atom  Na„SnBe (Si 0 ).2H 0,  i n t o account  narrow  of  135°  9  2  as w e l l . A l l of  of and  /(0-0-0 ) , are br  156° with an average between 140° and  3  /Si—O—Si  and  160° with an between  121°  149°.  V^ DISCUSSION OF ENERGY CALCULATIONS  1 . S i n g l e chain  To  select  through (*  selection  the  p o s s i b l e chains the computer program steps  values for predetermined  values of  (|>  and  0  .  The  14  Table I I I . L i s t of three-repeat chains composed of two f i x e d pairs of t e t r a h e d r a . The p a i r quoted i n parentheses represents the c l o s e s t approximation to the standard orientation of the third and f o u r t h t e t r a h e d r a . The / S i - O - S i vary between 120° and 180° with an average of 140°. The 7(0-0-0). vary between 90° and 180°. ~ b r  /(O-O-O)b r  /Si-O-Si  DT-Dz-CSt,)  145.00  150.00  122.53  144.47  147.09  143.78  140.00  138.82  142.17  142.18  143.56  145.00  131.57  117.22  144.87  150.27  123.08  139.47  123.19  122.62  121.16  128.37  109.47  140.92  140.00  152.32  135.74  119.33  161.33  140.00  133.01  144.87  119.99  149.78  145.00  130.67  144.87  176.61  148.27  145.00  130.96  117.22  176.61  119.44  155.53  135.00  159.47  155.53  125.00  166.87  137.06  164.47  147.32  Dj-St^-tD,) 135.00 150.00  132.23  139.11  150.36  166.83  151.47  1 32. 1 1  90.23  136.62  D -D -(D ) 2  2  2  140.00  D -D„-(D ) 2  2  145.00  D,-S -(St,) 2  150.00  145.00  D - S , - ( S )' 2  2  120.00  179.88  D -S -(St ) 2  2  3  130.00 D -S -(D ) 2  3  2  145.00 D -S„-(D 2  2  )  145.00 Da-S„-(D ) 2  145.00  D,-St,-(St,)  135.00  130.00  D,-St,-(S ) 2  130.00  D -St -(S ) 2  2  125.00  2  145.00  15  Table I I I . (continued)  /(o-o-o) .  /Si-O-Si  bf  D -St -(St ) 145.00 135.00  139.71  144.87  106.28  139.81  D -St,-(St ) 135.00 160.00  124.05  141.29  149.74  167.55  D -St -(S ) 150.00 135.00  135.94  147.09  148.50  132.66  D -St,-(St,) 135.00 150.00  137.79  135.00  91.18  133.12  D„-St5-(D„) 130.00 130.00  161.00  105.30  149.67  133.87  S -S -(St ) 140.00 140.00  140.53  119.13  119.13  146.45  S -S„-(D ) 150.00 150.00  122.67  172.26  127.15  134.16  S -St -(St ) 145.00 150.00  124.85  123.19  113.55  110.72  S -St -(D ) 135.00 165.00  119.97  115.00  167.03  115.56  5 - St -(S ) 140.00 140.00  141.06  119.13  146.88  120.00  5 - St,-(S ) 140.00 140.00  141.57  119.19  145.55  119.41  S,-St -(St ) 135.00 140.00  147.45  135.00  179.14  135.53  S,-St -(St,) 165.00 135.00  119.34  159.30  106.30  120.33  S -St -(D ) 135.00 165.00  120.14  174.75  167.03  161.73  S„-St -(St ) 135.00 140.00  143.93  174.75  146.88  144.04  2  3  1  2  5  2  5  2  3  2  2  5  2  2  2  2  3  3  ((  2  1  5  2  3  2  1  5  3  a  4  5  1  5  16  Table I I I . (continued)  /Si-O-Si  St , - s t  -(D ) 130.00  /(0-0-0)  2  155.00  134.47  155.00  130.00  155.00  122.92  130.00  107.46  144.61  147.45  135.00  108.94  153.93  137.79  135.00  154.10  147.82  142.06  135.00  146.88  167.23  133. 18  90.23  108.94  100.97  151.37  144.60  132.86  151.76  120.00  149.58  120.00  137.51  120.00  128.68  120.00  120.00  150.00  120.72  141.86  113.55  125.16  St,-St«-(St ) 130.00 170.00  121.50  137.06  171.35  145.34  St«-St -(D ) 125.00 145.00  151.37  132.86  144.60  151.73  St -St -(D ) 150.00 150.00  120.00  141.86  141.86  137.51  St ! - S t - -(D ) 2  2  130.00  165.00  S t , - S t - •(D ) 3  2  135.00  140.00  St ,-St«- •(D,)  135.00  150.00  St ,-St s- •(D ) 2  135.00  140.00  St - S t - •(S ) 2  2  3  145.00  140.00  St 2-St,- •(D ) 2  145.00  125.00  S t - S t - •(DJ 2  5  120.00  179.88  St - S t « - •(St, ) 3  1 20.00  179.88  St - S t - •(St,) 3  5  150.00  5  5  5  5  2  2  17 J.  Table IV. / S i - O - S i w o l l a s t o n i t e (W).  and  B/  /Si2-07-Si3 7si1-08-Si3 /Si1-09-Si2 AVE/Si-O-Si /09-07-08 709-08-07 707-09-08 AVE/(0-0-0)  b r  for  Ca-B  AVE/Si-O-Si /O9-07-08 709-08-07 707-09-08 AVE/ (0—0—0)  b f m  1  3  1  2  139. 1 7° 1 34. 95° 159. 91 °  1 37.50° * 1 34. 01 ° ** 1 62. 49°  1 44.06°  143.25°  1 44. 68°  1 44. 67°  1 53.33° 1 48.87° 1 22.74°  156.27° 150.17° 126.75°  1 54. 30° 1 48. 84° 123. 05°  1 55.15° 1 46. 37° 121 .94°  141 .65°  144.40°  1 42. 06°  141. 15°  Mn--W  Fe--W  1  and  Fe--B  138.15° 136.19° 155.47°  1  Para -W  1 39.24° 1 40.18° 1 49.14°  37° 1 39. 1 40. 08° 1 50. 54°  139. 18° 1 39.88° 151 .1 1 °  140. 24° 1 40. 42° 149. 33°  142 .85°  1 43 33° .  1 43. 39°  1 44. 33°  1 55.37° 1 56.67° 1 32.00°  1 55.02° 1 56. 37° 131. 41  1 54. 72° 1 56. 25° 1 30. 99°  1 56. 46° 1 55. 84° 1 32. 30°  1 48.01 °  147. 60°  147. 32°  1 48 20° .  * /Si1-07-Si3 ** /Si2-08-Si3 Osashi and F i n g e r (1978) Rapoport and Burnham (1973) Peacor and Prewitt (1963) • T r o j e r (1968) 2  Mn--B  (B).  1 35.59° 1 35.80° 160 .80°  wi  /Si1-07-Si3 7si2-08-Si3 /Si1-09-Si2  1  bustamite  0  18  steps  were  angular and  usually  values was  /(0—0—0)  b r %  5°  increments.  The  chains f o r each set of  then checked to determine that satisfied  the  previously  r e s t r i c t i o n s . Energy c a l c u l a t i o n s were then out of a p r o x i m a t i v e l y these their  the.  /Si—0—Si  mentioned  angular  undertaken  for  182,000 chains generated.  Over one  61  h a l f of  chains a l s o have the s m a l l e s t p o s s i b l e d i f f e r e n c e between i n d i v i d u a l /Si—O—Si  and  therefore  the  largest  average  d(Si-Si).  As  shown  on  represented symmetry  Figure  3  by three  an  segments  non—equivalent  tetrahedron was  i n f i n i t e three—repeat chain can each  neighbour variation  have  comparable  interactions  non—bridging /Si-O-H of  are  three  the  fourth  the  atomic  i . e . The  relatively  three  second and  small.  The  third  largest  segments of the same chain shown  %.  cases  hydrogen  oxygen atoms at a  180°. The  of  to a s c e r t a i n that these  energies;  in energy between two  all  However  i n c l u d e d to more f u l l y d e s c r i b e a l l  on F i g u r e 3 i s 0.001 In  consisting  tetrahedra.  i n t e r a c t i o n s . I t i s a l s o necessary segments  one  be  atoms  were  distance  resulting S i „ 0  1 3  H  1 0  of  appended 0.95  cluster  A  to  the  and  with  is neutral.  2. R e s u l t s of energy c a l c u l a t i o n s  It can be shown that for chains with average / S i — O - S i to  a  constant  those  chains  with  average have the l a r g e s t d ( S i — S i ) . The  all  equal  / S i — O — S i equal to the  total  energy  of  chains  Figure 3. Three segments of the same chain 7256.172, -256.170 and -256.170 A.U r e s p e c t i v e l y .  whose  energies are  20  can then be r e l a t e d to the / S i - O - S i through Figure  4  d(Si—Si)  in  -256.168  and  the d ( S i - S i ) .  shows a p l o t of t o t a l energy the  cluster.  -256.147  The  A.U.  total  While  a g a i n s t the average  energy  varies  the average  between  d(Si-Si)  from 2.93 to 3.04 A. A weak negative c o r r e l a t i o n e x i s t s these  two  as  indicated  regression c o e f f i c i e n t  by  this  population  approximatively  to  d(Si—Si).  shows  The  top  six row  and s i n g l e — e e l i p s e d  every  other  magnitude  as  the  significant  trend can be a  with  of  = .63  2  average  /Si—0—Si  than those  with  angle. plots  of  pairs.  discrepancy  segments  with  with  those with a l l three / S i — O — S i  the  total chains Since  on the v e r t i c a l  different  chains  140°,  includes  double  bar  chains  R  g e n e r a l l y have lower energy  / S i - O — S i c o v e r i n g a wider Figure 5  equation  (14) d ( S i - S i )  of  equal  c l o s e to the average  regression  between  0.63.  E = 267 (42) - 0.142 For  the  ranges  energy  vs  average  with combinations the  spacing  of  between  s c a l e i s of the same order of  between same  observed  calculations  chain;  from  the  no  on  the  statistically  diagram  for  double and a s i n g l e — e e l i p s e d p a i r or those  two s i n g l e — e e l i p s e d p a i r s . A l l diagrams i n the f i r s t  the with  row c o n t a i n  very few p o i n t s r e f l e c t i n g geometry c o n s t r a i n t s on these c h a i n s . The chains composed of those  composed  range of average double—eelipsed The  second  of  double  two  d(Si—Si)  and  single—eclipsed  single—eclipsed than  the  pairs  and  p a i r s have a narrower  chains  composed  of  two  pairs. row i n c l u d e s c h a i n s composed of combinations  of  ' 2-83  1  1  1 2.97  1  1  l 3.00  i  l  t 3.04  AVERAGE d(Si-Si) A  Figure 4. Total energy (A.U.) a g a i n s t a v e r a g e d ( S i - S i ) (A) , t h o s e c h a i n s w i t h t h e l a r g e s t d ( S i - S i ) have t h e l o w e s t e n e r g y . [ t h e s p a c i n g on t h e h o r i z o n t a l axis i s 0.012 A the inaccuracy reflects the round-off e r r o r ]  -8M.147  D-D  3  s-s  D-S  4  >. .IM  a  s s  3  o  .1*1  •  i  i i i i i  >  i  i »  i i t i  "S  i  i  i  i i i  i i »  •Ml 147  s-st  D-St  St-St  >. .164 O  a s  ^  •V .Ml 1  aas  .1H  MT  I  »  I  I  I  «JOO  AVEfUUM d(tt-H) A  AVERAOf 4UU-80 A  i  i  i  i i ur  AVERAOK dlH-SU A  Figure 5 . T o t a l energy (A.U.) a g a i n s t average d ( S i - S i ) (A) , f o r each group of c h a i n s . [ the spacing on the h o r i z o n t a l a x i s i s 0 . 0 1 2 A the inaccuracy r e f l e c t s the round-off e r r o r ]  23  one staggered p a i r with e i t h e r a  single  or  a double—eclipsed  p a i r or another staggered p a i r . A l l of these combinations have a larger  number of c h a i n s . The f i r s t  two diagrams  show a range of  d ( S i — S i ) between 2.97 and 3.03 A. However i n the f i r s t energy  varies  second  case  Although  between  -256.163 and -256.147 A.U. While  i t varies  chains  case  between  consisting  of  -256.168 a  and  the  i n the  -256.147  A.U.  staggered p a i r and e i t h e r a  double or a s i n g l e — e c l i p s e d p a i r cover the same range of average d ( S i — S i ) the l a t t e r group has c h a i n s with both  cases  a  lower  energy.  the c h a i n s with the h i g h e s t energy has one / S i — O — S i  l e s s than  110°. The  d(Si—Si)  between  last 2.93  diagram and  shows  a  range  of  average  3.03 A with energy v a r y i n g between  -256.164 and -256.152 A.U. The two c h a i n s with the h i g e s t are  those with two / S i — O — S i equal to  range  of  In  average  d(Si—Si)  is  120°.  wider  In  than  this in  energy  case  the  the  previous  examples. Although a combination of s i n g l e and staggered p a i r s chains with the lowest energy  yield  i . e . -256.167 A.U. Any combination  i n c l u d i n g at l e a s t a staggered p a i r of t e t r a h e d r a tend to have a lower energy  Figure  than a chain c o n t a i n i n g no such p a i r .  6  shows  the  generated which i s c l o s e s t illustrate  wollastonite in  chain  comformation.  the chains with the lowest energy  groups of t h e o r e t i c a l c h a i n s  3. Packing of chains  and  Figures  the 7  chain and  8  i n each of the s i x  WOLLASTONITE  - D - St 2  4  Figure 6. W o l l a s t o n i t e (Buerger and P r e w i t t , 1 9 6 1 ) and i t s c l o s e s t a p p r o x i m a t i o n w i t h t o t a l e n e r g y -256.158 A.U.  D -D -D 2  2  2  D -S -St 1  2  4  S -S -St 2  2  5  F i g u r e 7. C h a i n s c o n t a i n i n g a c o m b i n a t i o n o f d o u b l e o r s i n g l e - e c l i p s e d p a x r s wath e n e r g i e s -256.164, -256.163 a n d -256.163 A.U. r e s p e c t i v e l y .  D - St - St., 2  3  S -St -D 2  4  2  St -St -D 1  3  2  F i g u r e 8. C h a i n s c o n t a i n i n g a s t a g g e r e d p a i r o f t e t r a h e d r a w i t h e i t h e r a double or s i n g l e e c l i p s e d or another staggered pair. Energies are - 2 5 6 . 1 6 5 , -256.167 a n d -256.164 A.U. r e s p e c t i v e l y .  27  Liebau  (1980) d e f i n e d the s t r e t c h i n g C ^  f a c t o r of a chain as  A M I ,  I f  = 1 c  i s the length of the c h a i n , I t  where I the  tetrahedral  f a c t o r of odd chains.  It  the r a d i i energy gives  P  edge  i s the  and p i s the p e r i o d i c i t y . The  repeat chains i s l a r g e r than that of appears  length  stretching  even  repeat  to be r e l a t e d to the e l e c t r o n e g a t i v i t y  total  a g a i n s t the s t r e t c h i n g f a c t o r for the t h e o r e t i c a l  chains  regression  correlation  exist  i.e.,  chain to have  a  generated  this  in  coefficient  lower  regression  0.01  Of  those  study  low  only the S —St„ —(D ) 2  chains appear u n l i k e l y to lend themselves to further  showing  that  no  there i s no tendency for a s t r a i g h t e r  energy.  study  of  A  and  of  a  of the i n t e r s t i t i a l c a t i o n s .  of  2  energy  chains  and S — S t , — ( D ) 3  close  2  packing.  i s r e q u i r e d to determine i f the remaining  can be packed such that i n t e r s t i t i a l c a t i o n s can be in a c r y s t a l c h e m i c a l l y reasonnable  A  chains  coordinated  manner.  VI. CONCLUSION  The two  total  energy  of three—repeat chains i s i n f l u e n c e d by  f a c t o r s . F i r s t , chains with the l a r g e s t d ( S i - S i ) ,  with a l l three / S i — O - S i equal to 140° have Secondly,  chains  a l s o have a lower  with  the  i . e , those  lowest  energy.  at l e a s t a staggered p a i r of t e t r a h e d r a  energy than those with no such p a i r .  This  is  28  in  keeping  with  the.  f a c t that f o r p a i r s of t e t r a h e d r a with a  constant / S i — O — S i the staggered conformation has a lower  energy  than  It  either  noteworthy  the  that the two  sodium s i l i c a t e tetrahedra of  single  or  double—eclipsed  lowest energy two—repeat  pairs.  c h a i n s are  is the  type which i s composed of two p a i r s of staggered  and the pyroxene  type which i s composed of two  pairs  single—eclipsed tetrahedra. Under the geometric r e s t r i c t i o n s of t h i s study there  three—repeat  chains d i f f e r e n t  exist  from the one type found i n nature  and the energy of a number of these c h a i n s i s as  low  or  lower  than that of the c l o s e s t approximation to the w o l l a s t o n i t e type. In  the group of two—repeat  to pack i n such a way  c h a i n s those of lowest energy happen  as to accommodate i n t e r s t i t i a l  the three—repeat chains there chains sites  which for  necessary average  cannot  determine  d(Si—Si)  interstitial  to  be  many  low  energy  pack together such as to p r o v i d e s u i t a b l e  interstitial to  appear  c a t i o n s . In  and  cations.  Further  is  i f i t i s p o s s i b l e to optimize both the the  bonding  requirements  c a t i o n s f o r chains d i f f e r e n t  type. No c o r r e l e t i o n was  investigation  s t r a i g h t e r chains to have lower  indicating  energy.  the  from the w o l l a s t o n i t e  found between the s t r e t c h i n g  the energy of three—repeat c h a i n s  of  no  factor  tendency  and for  29  REFERENCES  Buerger, M.J and C.T. P r e w i t t , 1961, The c r y s t a l s t r u c t u r e of w o l l a s t o n i t e and p e c t o l i t e ; Proc N.A.S 47,1884-1888. Dobash, P.A, 1974, Quantum Chemistry (available from QCEP, Indiana Indiana 47410).  Program Exchange No. 141 University, Bloomington,  Dewar, M.J.S., 1969, The molecular o r b i t a l theory of organic chemistry: McGraw-Hill. Fock, V., 1930, Naherungsmethode zur Lozung des quantenmechanischen Mehrkorperproblems: Z. 126-146  Physik,  Hartree, D.R., 1927, The wave mechanics of an atom with a coulomb c e n t r a l f i e l d Part I . Theory and methods: Cambridge P h i l . S o c , 24, 89-110.  61,  nonProc.  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